How to alter woocommerce widget filter by attribute widget?

What about extending its class to override the widget?

in your functions.php, add this class, you can then change the elements or do conditional statements depending on what you need.

/**
 * Layered nav widget
 *
 * @package WooCommerce/Widgets
 * @version 2.6.0
 */

/**
 * Widget layered nav class.
 */
class The_New_Attribute_Widget extends WC_Widget_Layered_Nav {

    /**
     * Constructor.
     */
    public function __construct() {
        $this->widget_cssclass    = 'woocommerce widget_layered_nav woocommerce-widget-layered-nav';
        $this->widget_description = __( 'Display a list of attributes to filter products in your store.', 'woocommerce' );
        $this->widget_id          = 'woocommerce_layered_nav';
        $this->widget_name        = __( 'Filter Products by Attribute', 'woocommerce' );
        parent::__construct();
    }

    /**
     * Updates a particular instance of a widget.
     *
     * @see WP_Widget->update
     *
     * @param array $new_instance New Instance.
     * @param array $old_instance Old Instance.
     *
     * @return array
     */
    public function update( $new_instance, $old_instance ) {
        $this->init_settings();
        return parent::update( $new_instance, $old_instance );
    }

    /**
     * Outputs the settings update form.
     *
     * @see WP_Widget->form
     *
     * @param array $instance Instance.
     */
    public function form( $instance ) {
        $this->init_settings();
        parent::form( $instance );
    }

    /**
     * Init settings after post types are registered.
     */
    public function init_settings() {
        $attribute_array      = array();
        $std_attribute        = '';
        $attribute_taxonomies = wc_get_attribute_taxonomies();

        if ( ! empty( $attribute_taxonomies ) ) {
            foreach ( $attribute_taxonomies as $tax ) {
                if ( taxonomy_exists( wc_attribute_taxonomy_name( $tax->attribute_name ) ) ) {
                    $attribute_array[ $tax->attribute_name ] = $tax->attribute_name;
                }
            }
            $std_attribute = current( $attribute_array );
        }

        $this->settings = array(
            'title'        => array(
                'type'  => 'text',
                'std'   => __( 'Filter by', 'woocommerce' ),
                'label' => __( 'Title', 'woocommerce' ),
            ),
            'attribute'    => array(
                'type'    => 'select',
                'std'     => $std_attribute,
                'label'   => __( 'Attribute', 'woocommerce' ),
                'options' => $attribute_array,
            ),
            'display_type' => array(
                'type'    => 'select',
                'std'     => 'list',
                'label'   => __( 'Display type', 'woocommerce' ),
                'options' => array(
                    'list'     => __( 'List', 'woocommerce' ),
                    'dropdown' => __( 'Dropdown', 'woocommerce' ),
                ),
            ),
            'query_type'   => array(
                'type'    => 'select',
                'std'     => 'and',
                'label'   => __( 'Query type', 'woocommerce' ),
                'options' => array(
                    'and' => __( 'AND', 'woocommerce' ),
                    'or'  => __( 'OR', 'woocommerce' ),
                ),
            ),
        );
    }

    /**
     * Get this widgets taxonomy.
     *
     * @param array $instance Array of instance options.
     * @return string
     */
    protected function get_instance_taxonomy( $instance ) {
        if ( isset( $instance['attribute'] ) ) {
            return wc_attribute_taxonomy_name( $instance['attribute'] );
        }

        $attribute_taxonomies = wc_get_attribute_taxonomies();

        if ( ! empty( $attribute_taxonomies ) ) {
            foreach ( $attribute_taxonomies as $tax ) {
                if ( taxonomy_exists( wc_attribute_taxonomy_name( $tax->attribute_name ) ) ) {
                    return wc_attribute_taxonomy_name( $tax->attribute_name );
                }
            }
        }

        return '';
    }

    /**
     * Get this widgets query type.
     *
     * @param array $instance Array of instance options.
     * @return string
     */
    protected function get_instance_query_type( $instance ) {
        return isset( $instance['query_type'] ) ? $instance['query_type'] : 'and';
    }

    /**
     * Get this widgets display type.
     *
     * @param array $instance Array of instance options.
     * @return string
     */
    protected function get_instance_display_type( $instance ) {
        return isset( $instance['display_type'] ) ? $instance['display_type'] : 'list';
    }

    /**
     * Output widget.
     *
     * @see WP_Widget
     *
     * @param array $args Arguments.
     * @param array $instance Instance.
     */
    public function widget( $args, $instance ) {
        if ( ! is_shop() && ! is_product_taxonomy() ) {
            return;
        }

        $_chosen_attributes = WC_Query::get_layered_nav_chosen_attributes();
        $taxonomy           = $this->get_instance_taxonomy( $instance );
        $query_type         = $this->get_instance_query_type( $instance );
        $display_type       = $this->get_instance_display_type( $instance );

        if ( ! taxonomy_exists( $taxonomy ) ) {
            return;
        }

        $terms = get_terms( $taxonomy, array( 'hide_empty' => '1' ) );

        if ( 0 === count( $terms ) ) {
            return;
        }

        ob_start();

        $this->widget_start( $args, $instance );

        if ( 'dropdown' === $display_type ) {
            wp_enqueue_script( 'selectWoo' );
            wp_enqueue_style( 'select2' );
            $found = $this->layered_nav_dropdown( $terms, $taxonomy, $query_type );
        } else {
            $found = $this->layered_nav_list( $terms, $taxonomy, $query_type );
        }

        $this->widget_end( $args );

        // Force found when option is selected - do not force found on taxonomy attributes.
        if ( ! is_tax() && is_array( $_chosen_attributes ) && array_key_exists( $taxonomy, $_chosen_attributes ) ) {
            $found = true;
        }

        if ( ! $found ) {
            ob_end_clean();
        } else {
            echo ob_get_clean(); // @codingStandardsIgnoreLine
        }
    }

    /**
     * Return the currently viewed taxonomy name.
     *
     * @return string
     */
    protected function get_current_taxonomy() {
        return is_tax() ? get_queried_object()->taxonomy : '';
    }

    /**
     * Return the currently viewed term ID.
     *
     * @return int
     */
    protected function get_current_term_id() {
        return absint( is_tax() ? get_queried_object()->term_id : 0 );
    }

    /**
     * Return the currently viewed term slug.
     *
     * @return int
     */
    protected function get_current_term_slug() {
        return absint( is_tax() ? get_queried_object()->slug : 0 );
    }

    /**
     * Show dropdown layered nav.
     *
     * @param  array  $terms Terms.
     * @param  string $taxonomy Taxonomy.
     * @param  string $query_type Query Type.
     * @return bool Will nav display?
     */
    protected function layered_nav_dropdown( $terms, $taxonomy, $query_type ) {
        global $wp;
        $found = false;

        if ( $taxonomy !== $this->get_current_taxonomy() ) {
            $term_counts          = $this->get_filtered_term_product_counts( wp_list_pluck( $terms, 'term_id' ), $taxonomy, $query_type );
            $_chosen_attributes   = WC_Query::get_layered_nav_chosen_attributes();
            $taxonomy_filter_name = wc_attribute_taxonomy_slug( $taxonomy );
            $taxonomy_label       = wc_attribute_label( $taxonomy );

            /* translators: %s: taxonomy name */
            $any_label      = apply_filters( 'woocommerce_layered_nav_any_label', sprintf( __( 'Any %s', 'woocommerce' ), $taxonomy_label ), $taxonomy_label, $taxonomy );
            $multiple       = 'or' === $query_type;
            $current_values = isset( $_chosen_attributes[ $taxonomy ]['terms'] ) ? $_chosen_attributes[ $taxonomy ]['terms'] : array();

            if ( '' === get_option( 'permalink_structure' ) ) {
                $form_action = remove_query_arg( array( 'page', 'paged' ), add_query_arg( $wp->query_string, '', home_url( $wp->request ) ) );
            } else {
                $form_action = preg_replace( '%/page/[0-9]+%', '', home_url( trailingslashit( $wp->request ) ) );
            }

            echo '<form method="get" action="' . esc_url( $form_action ) . '" class="woocommerce-widget-layered-nav-dropdown">';
            echo '<select class="woocommerce-widget-layered-nav-dropdown dropdown_layered_nav_' . esc_attr( $taxonomy_filter_name ) . '"' . ( $multiple ? 'multiple="multiple"' : '' ) . '>';
            echo '<option value="">' . esc_html( $any_label ) . '</option>';

            foreach ( $terms as $term ) {

                // If on a term page, skip that term in widget list.
                if ( $term->term_id === $this->get_current_term_id() ) {
                    continue;
                }

                // Get count based on current view.
                $option_is_set = in_array( $term->slug, $current_values, true );
                $count         = isset( $term_counts[ $term->term_id ] ) ? $term_counts[ $term->term_id ] : 0;

                // Only show options with count > 0.
                if ( 0 < $count ) {
                    $found = true;
                } elseif ( 0 === $count && ! $option_is_set ) {
                    continue;
                }

                echo '<option value="' . esc_attr( urldecode( $term->slug ) ) . '" ' . selected( $option_is_set, true, false ) . '>' . esc_html( $term->name ) . '</option>';
            }

            echo '</select>';

            if ( $multiple ) {
                echo '<button class="woocommerce-widget-layered-nav-dropdown__submit" type="submit" value="' . esc_attr__( 'Apply', 'woocommerce' ) . '">' . esc_html__( 'Apply', 'woocommerce' ) . '</button>';
            }

            if ( 'or' === $query_type ) {
                echo '<input type="hidden" name="query_type_' . esc_attr( $taxonomy_filter_name ) . '" value="or" />';
            }

            echo '<input type="hidden" name="filter_' . esc_attr( $taxonomy_filter_name ) . '" value="' . esc_attr( implode( ',', $current_values ) ) . '" />';
            echo wc_query_string_form_fields( null, array( 'filter_' . $taxonomy_filter_name, 'query_type_' . $taxonomy_filter_name ), '', true ); // @codingStandardsIgnoreLine
            echo '</form>';

            wc_enqueue_js(
                "
                // Update value on change.
                jQuery( '.dropdown_layered_nav_" . esc_js( $taxonomy_filter_name ) . "' ).change( function() {
                    var slug = jQuery( this ).val();
                    jQuery( ':input[name="filter_" . esc_js( $taxonomy_filter_name ) . ""]' ).val( slug );

                    // Submit form on change if standard dropdown.
                    if ( ! jQuery( this ).attr( 'multiple' ) ) {
                        jQuery( this ).closest( 'form' ).submit();
                    }
                });

                // Use Select2 enhancement if possible
                if ( jQuery().selectWoo ) {
                    var wc_layered_nav_select = function() {
                        jQuery( '.dropdown_layered_nav_" . esc_js( $taxonomy_filter_name ) . "' ).selectWoo( {
                            placeholder: decodeURIComponent('" . rawurlencode( (string) wp_specialchars_decode( $any_label ) ) . "'),
                            minimumResultsForSearch: 5,
                            width: '100%',
                            allowClear: " . ( $multiple ? 'false' : 'true' ) . ",
                            language: {
                                noResults: function() {
                                    return '" . esc_js( _x( 'No matches found', 'enhanced select', 'woocommerce' ) ) . "';
                                }
                            }
                        } );
                    };
                    wc_layered_nav_select();
                }
            "
            );
        }

        return $found;
    }

    /**
     * Count products within certain terms, taking the main WP query into consideration.
     *
     * This query allows counts to be generated based on the viewed products, not all products.
     *
     * @param  array  $term_ids Term IDs.
     * @param  string $taxonomy Taxonomy.
     * @param  string $query_type Query Type.
     * @return array
     */
    protected function get_filtered_term_product_counts( $term_ids, $taxonomy, $query_type ) {
        global $wpdb;

        $tax_query  = WC_Query::get_main_tax_query();
        $meta_query = WC_Query::get_main_meta_query();

        if ( 'or' === $query_type ) {
            foreach ( $tax_query as $key => $query ) {
                if ( is_array( $query ) && $taxonomy === $query['taxonomy'] ) {
                    unset( $tax_query[ $key ] );
                }
            }
        }

        $meta_query     = new WP_Meta_Query( $meta_query );
        $tax_query      = new WP_Tax_Query( $tax_query );
        $meta_query_sql = $meta_query->get_sql( 'post', $wpdb->posts, 'ID' );
        $tax_query_sql  = $tax_query->get_sql( $wpdb->posts, 'ID' );

        // Generate query.
        $query           = array();
        $query['select'] = "SELECT COUNT( DISTINCT {$wpdb->posts}.ID ) as term_count, terms.term_id as term_count_id";
        $query['from']   = "FROM {$wpdb->posts}";
        $query['join']   = "
            INNER JOIN {$wpdb->term_relationships} AS term_relationships ON {$wpdb->posts}.ID = term_relationships.object_id
            INNER JOIN {$wpdb->term_taxonomy} AS term_taxonomy USING( term_taxonomy_id )
            INNER JOIN {$wpdb->terms} AS terms USING( term_id )
            " . $tax_query_sql['join'] . $meta_query_sql['join'];

        $query['where'] = "
            WHERE {$wpdb->posts}.post_type IN ( 'product' )
            AND {$wpdb->posts}.post_status = 'publish'"
            . $tax_query_sql['where'] . $meta_query_sql['where'] .
            'AND terms.term_id IN (' . implode( ',', array_map( 'absint', $term_ids ) ) . ')';

        $search = WC_Query::get_main_search_query_sql();
        if ( $search ) {
            $query['where'] .= ' AND ' . $search;
        }

        $query['group_by'] = 'GROUP BY terms.term_id';
        $query             = apply_filters( 'woocommerce_get_filtered_term_product_counts_query', $query );
        $query             = implode( ' ', $query );

        // We have a query - let's see if cached results of this query already exist.
        $query_hash = md5( $query );

        // Maybe store a transient of the count values.
        $cache = apply_filters( 'woocommerce_layered_nav_count_maybe_cache', true );
        if ( true === $cache ) {
            $cached_counts = (array) get_transient( 'wc_layered_nav_counts_' . sanitize_title( $taxonomy ) );
        } else {
            $cached_counts = array();
        }

        if ( ! isset( $cached_counts[ $query_hash ] ) ) {
            $results                      = $wpdb->get_results( $query, ARRAY_A ); // @codingStandardsIgnoreLine
            $counts                       = array_map( 'absint', wp_list_pluck( $results, 'term_count', 'term_count_id' ) );
            $cached_counts[ $query_hash ] = $counts;
            if ( true === $cache ) {
                set_transient( 'wc_layered_nav_counts_' . sanitize_title( $taxonomy ), $cached_counts, DAY_IN_SECONDS );
            }
        }

        return array_map( 'absint', (array) $cached_counts[ $query_hash ] );
    }

    /**
     * Show list based layered nav.
     *
     * @param  array  $terms Terms.
     * @param  string $taxonomy Taxonomy.
     * @param  string $query_type Query Type.
     * @return bool   Will nav display?
     */
    protected function layered_nav_list( $terms, $taxonomy, $query_type ) {
        // List display.
        echo '<ul class="woocommerce-widget-layered-nav-list">';

        $term_counts        = $this->get_filtered_term_product_counts( wp_list_pluck( $terms, 'term_id' ), $taxonomy, $query_type );
        $_chosen_attributes = WC_Query::get_layered_nav_chosen_attributes();
        $found              = false;
        $base_link          = $this->get_current_page_url();

        foreach ( $terms as $term ) {
            $current_values = isset( $_chosen_attributes[ $taxonomy ]['terms'] ) ? $_chosen_attributes[ $taxonomy ]['terms'] : array();
            $option_is_set  = in_array( $term->slug, $current_values, true );
            $count          = isset( $term_counts[ $term->term_id ] ) ? $term_counts[ $term->term_id ] : 0;

            // Skip the term for the current archive.
            if ( $this->get_current_term_id() === $term->term_id ) {
                continue;
            }

            // Only show options with count > 0.
            if ( 0 < $count ) {
                $found = true;
            } elseif ( 0 === $count && ! $option_is_set ) {
                continue;
            }

            $filter_name    = 'filter_' . wc_attribute_taxonomy_slug( $taxonomy );
            $current_filter = isset( $_GET[ $filter_name ] ) ? explode( ',', wc_clean( wp_unslash( $_GET[ $filter_name ] ) ) ) : array(); // WPCS: input var ok, CSRF ok.
            $current_filter = array_map( 'sanitize_title', $current_filter );

            if ( ! in_array( $term->slug, $current_filter, true ) ) {
                $current_filter[] = $term->slug;
            }

            $link = remove_query_arg( $filter_name, $base_link );

            // Add current filters to URL.
            foreach ( $current_filter as $key => $value ) {
                // Exclude query arg for current term archive term.
                if ( $value === $this->get_current_term_slug() ) {
                    unset( $current_filter[ $key ] );
                }

                // Exclude self so filter can be unset on click.
                if ( $option_is_set && $value === $term->slug ) {
                    unset( $current_filter[ $key ] );
                }
            }

            if ( ! empty( $current_filter ) ) {
                asort( $current_filter );
                $link = add_query_arg( $filter_name, implode( ',', $current_filter ), $link );

                // Add Query type Arg to URL.
                if ( 'or' === $query_type && ! ( 1 === count( $current_filter ) && $option_is_set ) ) {
                    $link = add_query_arg( 'query_type_' . wc_attribute_taxonomy_slug( $taxonomy ), 'or', $link );
                }
                $link = str_replace( '%2C', ',', $link );
            }

            if ( $count > 0 || $option_is_set ) {
                $link      = apply_filters( 'woocommerce_layered_nav_link', $link, $term, $taxonomy );
                $term_html = '<a rel="nofollow" href="' . esc_url( $link ) . '">' . esc_html( $term->name ) . '</a>';
            } else {
                $link      = false;
                $term_html = '<span>' . esc_html( $term->name ) . '</span>';
            }

            $term_html .= ' ' . apply_filters( 'woocommerce_layered_nav_count', '<span class="count">(' . absint( $count ) . ')</span>', $count, $term );

            echo '<li class="woocommerce-widget-layered-nav-list__item wc-layered-nav-term ' . ( $option_is_set ? 'woocommerce-widget-layered-nav-list__item--chosen chosen' : '' ) . '">';
            echo apply_filters( 'woocommerce_layered_nav_term_html', $term_html, $term, $link, $count ); // WPCS: XSS ok.
            echo '</li>';
        }

        echo '</ul>';

        return $found;
    }
}

then in your functions.php again, unregister the default attribute widget and register the new one above:

function your_widget_handler() {
      unregister_widget( 'WC_Widget_Layered_Nav' ); //unregister
      register_widget( 'The_New_Attribute_Widget' ); //Register the new class
} 
add_action( 'widgets_init', 'your_widget_handler' );

I took a look at the source code and it seems that it's only possible if you chose "List" as the "Display type". If so, then there is a filter applied to the displayed elements:

apply_filters( 'woocommerce_layered_nav_term_html', $term_html, $term, $link, $count );

so in order to add custom filter, you can do:

function so_62519320_prepend_name( $term_html, $term, $link, $count ) {
    // do stuff
    return $some_modified_value;
}
add_filter( 'woocommerce_layered_nav_term_html', 'so_62519320_prepend_name', 10, 4 );

Unfortunately, if you chose "Dropdown" as the "Display type", then there are no filters applied and it seems that you can't modify labels. There's only

esc_html( $term->name )

used to display option labels, so the only way to change this is to change the value in the database.