Sometimes people, definitely not me, make typos when uploading avatars and would like the ability to change the name of their avatars using the website. There has also been a bug lately that replaces the avatar name with 'Loading...' even though a name was definitely added, seen as you can't upload without one.
Unless I'm mistaken, the only way to change an avatar name currently is by editing it in unity using the control panel. This can be a time consuming task if you've already closed unity, or worse, deleted the project T_T.
You already have the ability to amend Private/Public and to delete the avatar. Just add a teeny tiny pencil button next to the name. Kthanksbye.