How to update ShqrePoint Group name?

You could try to use js to test the request,after the test is successful,you fill in the value one by one.

This code works well:

<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.2/jquery.min.js"></script>

<script>
$(function(){
$("#btnClick").click(function(){
var requestUri = _spPageContextInfo.webAbsoluteUrl + "/_api/web/sitegroups/getbyId(23)";
$.ajax({
url: requestUri,
type: "POST",
data:JSON.stringify({'__metadata': { 'type': 'SP.Group' }, 'Title':'testing'}),
headers: {
"content-type": "application/json;odata=verbose",
"X-HTTP-Method":"MERGE",
"X-RequestDigest": $("#__REQUESTDIGEST").val()
},
success: onSuccess,
error: onError
});
function onSuccess(data) {
alert('Group description updated successfully');
}
function onError(error) {
alert(JSON.stringify(error));
}
});
});
</script>
<input type="button" id="btnClick" value="Update SharePoint Group Details"/>

Then you could find you need to add "X-HTTP-Method":"MERGE" in header.