How to change a users name on a Shared Photostream?
Following these steps should do the trick:


Unsubscribe to the shared photo album.
Change the full name on the Apple ID account.
Subscribe again to the Photo Stream by getting a new invitation.


Afterwards, all the comments you'll post should display the new name. The thing is that once a comment is posted, the name does not change, so if you want to change the name on an already posted comment, you'll have to Delete the comment, follow the steps, and post it again.