Unfortunately with how WordPress treats usernames there isn't the facility built in to change the username once its set - you can do it if you directly modify the database and change the user's name, but its not advisable if you're not comfortable with doing that, and it doesn't really let the user pick there own name, which is what you want.
The only real way for a user to pick their own username is if they register themselves, which isn't the scope of the plugin at this time I'm afraid.
There is, however, the facility for users to change their nickname, and then set that nickname as their 'display name' which should hide the username from being seen by others on the site. You would simply have to advise users you add to do this if they wish to hide their username from others. You could probably add this advice into the email it sends to newly registered users.
Hope that helps!