It's a simple task!
1. Click on your user settings tab
2. In the Account tab of the User Settings menu, enter your new name in the name slot:
3. Enter your password to show Discord you mean business:
4. Show Discord you're ready for commitment to your changes, by hitting the done button:
And you're all set! Keep in mind that usernames aren't unique, so you do technically have the ability to clone your friends' names. Remember, with great power comes great responsibility.. And it's probably best to not piss off your server mods.
Important note: Username changes are limited to 2 an hour.