To change your Twitch user name on the site, you will need to locate the settings options while you are logged in.
Start by punching all of your information, username, and password, and log into the site at Twitch.com. After you log in, locate the settings tab by clicking on your profile icon at the top right of your screen.
Doing so will open a properties tab that will display several options for your Twitch account. However, you will need to find the gear icon called settings, as shown below.
Under settings, scroll down until you see both Username and Displayname, as shown below. Username will change your account's current name that displays to all other Twitch users when you either comment in the chat of a stream or stream yourself.
Displayname is just an option to change the capitalization, so if you want to change your name on Twitch, you will need to change your username. That said, you can change your username by clicking the pencil icon to the right of the username box.
Doing so will take you to a tab, as shown above. Type whatever name you want instead of your current one inside the space above and hit save. If another user does not take the name, you will change your Twitch name after a few seconds.