Why do we spell the word “who” with a silent “w” when it isn’t needed?

There is no point in trying to make sense of English spelling. The reasons are often lost in history. English has undergone many transformations. Here is probably the best explanation you will get.

who (pron.) Old English hwa "who," sometimes "what; anyone, someone; each; whosoever," from Proto-Germanic *hwas (source also of Old Saxon hwe, Danish hvo, Swedish vem, Old Frisian hwa, Dutch wie, Old High German hwer, German wer, Gothic hvo (fem.) "who"), from PIE root *kwo-, stem of relative and interrogative pronouns. https://www.etymonline.com/word/who#etymonline_v_7978

The spelling "who" was originally used simply because, in past time periods, this word was pronounced with a "wh" sound. To be clear, by "a "wh" sound" I mean a sound that is different from either "w" or "h" on its own. Although many varieties of English have lost "wh" as a distinct sound, replacing it with plain [w], the digraph "wh" used to (and in some accents still does) have a value that can be transcribed as /hw/, or written with the special symbol [ʍ] if we interpret it as a single sound.

Many aspects of modern English spelling derive from Middle English pronunciation. Analogy with the spelling of other words probably contributed to the continued use of "wh" in spelling even after people started pronouncing the word with a simple /h/ sound. "Who" is one of the many question words starting with wh- in English.

Here is a brief explanation of why who lost its /w/ sound. The sound [w] is made by rounding the lips. The pronunciation of the word "who" evolved over time to have a rounded vowel sound, and the presence of a rounded vowel sound caused people to stop hearing a separate rounded consonant at the start of the word. This has happened in other words, most obviously two.

The question word how lost its /w/ sound for a similar reason, but much earlier, so it has a spelling without "wh".

Unfortunately, irregular spellings in English aren't always etymological. Because of the confusion between "h" and "wh" before rounded vowels, a few words pronounced with a simple /h/ sound are spelled with "wh" before "o" even though, from an etymological perspective, they would be expected to just have "ho". An example is whole, which is etymologically related to heal.

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A few blog posts that relate to topics brought up in the comments:

• "Hwæt about WH?", by Mark Liberman, 2011 April 13

• "The Secret Ways of Weak Forms: Here Comes a New ’Un", by Piotr Gąsiorowski, 2013 February 07

Why is the W silent in the word Who?

It's a long story (as @chaslyfromUK points out, it comes from a PIE root, which means it's been around on people's lips for over 4000 years), but the short of it is that Germanic languages like English changed /kʷ/ in PIE words to /hʷ/ or /xʷ/. English words beginning with WH, which used to be spelled HW, also used to be pronounced with much more friction than today -- like CH, the ach-Laut, the sound in the German word ach. English words like whale and wail did not start with the same consonant sound then -- today, for most Americans and some UK English speakers, they do.

The H part of the English WH words went away in most cases, like where, when, why, whether, which, what, whence, and whither; that was part of the change from Middle to Modern English. The old /x/ sounds that were allophones of /h/ went silent; they were spelled GH and our Middle English spelling system keeps them for luck or something.

In the cases of how and who, however, it was the W part of WH that went away. These two Middle English words contained back rounded vowels /u:/ in how and /o:/ in who, and if you've ever tried to pronounce words like woo or woad carefully you know that it's very hard to make the W part distinctive. This is because the sound /w/ is made simply by rounding the lips tightly, and they are already tightly rounded in making the vowels /u:/ and /o:/.

A consonant that can't be heard is often not written, and after a while nobody bothers.
That's all, really.

One of the oldest surviving works in any form of English is Beowulf. Here is the first page of the manuscript:

The first line is:

Hƿæt Ƿe garde

which in more modern letters is:

Hwæt We garde

since the ancient letter wynn (Ƿ ƿ) is now represented by double-u (W w).

The first word is "hwæt", which is related to the modern "what" and would have been pronounced "hwat" (to rhyme with "at"). In Middle English, we switched the "h" and "w" around in our spelling, although many speakers continued to pronounce the initial /h/ well into the 20th century.

"Who" is one of the "six Ws" question words:

• who /huː/
• what /wɒt/
• where /wɛə(ɹ)/
• when /wɛn/
• why /waɪ/
• how /haʊ/

For each of the above words starting with /w/, there are some people who still pronounce them starting with /ʍ/ (i.e. /hw/) instead, although this is becoming increasingly rare. All of these words at one point were pronounced starting with /ʍ/.

Wiktionary summarises how "who" became /huː/ nicely:

From [...] Old English hwā [...] Spelling change hw > wh in Middle English (without sound change in initial consonant cluster), while sound change /hw/ > /h/ due to wh-cluster reduction after a bizarre instance of three consecutive vowel shifts of /aː/ to /ɔː/ to /oː/ in Middle English and further to /uː/ in Early Modern English (although only the shift from /ɔː/ to /oː/ is unusual). Compare how, which underwent this change earlier (in Old English), and thus is spelt h.