"Once you've changed your name, the abandoned username will be held by Twitch for a minimum of six months," Twitch's Emily Halpin writes on the company blog. "After which, we may decide to return it to the pool of available usernames for the community and new users."
Halpin also says that some streamers will lose access to their streamer stats after changing their names. To sidestep that, you'l have to export your stats ahead of time.
The whole process sounds simple enough. Just hit your profile settings page and go from there. See PlayStation? This is how simple it could be to let everyone change their PlayStation Network handles.