Twitch has made good on its community's most frequent request: Changing one's username. Whether the original one you chose now sucks, or you want to make it consistent with a gamertag on some other service, the choice is yours.
The feature was announced yesterday afternoon and immediately went live. The option is found in the Profile Settings page of one's Twitch account. An option to change the username is highlighted beside the existing one.
That said, streamers who are concerned about maintaining their statistics should be sure export them prior to changing the name. The Twitch blog noted that in testing out the feature, Twitch discovered that some streamers lost access to their stats. Twitch also pointed out that outdated URLs will not redirect to new account names. So anyone advertising their page externally should be sure to update that.
Abandoned names will be held for at least six months and may be returned to the Twitch community for new use, but this is not automatic.
Twitch joins major gaming service providers like Steam and Xbox Live in letting users change their username, or at least the handle they present to others. (Xbox Live charges a fee for this.) Not on that list, after more than a decade, is PlayStation Network, where you're still stuck with whatever AOL Instant Messenger handle you were using in 2006 (like me).