We get a few of these every day. Without this facility we have to organise for members to contact a member of staff or post a message in the forum. An admin then has to edit the user to change their name. Sometimes they request a name which is already taken or not suitable. We then have to bounce messages back and forth until they pick an available/appropriate name.
Finally we change the name.
Alternatively a facility to request a new login name in a member's control panel makes the process much easier. Only valid login names are accepted.
The name change is put into a moderation queue and super moderators can approve or deny the name change.
This is available in vBulletin via a plugin:
It many not be something which is used much in other forums, but this functionality saves time and saving your time and that of your moderators is something we should always strive to achieve.