You'll need to complete the "Make a Name for Yourself" story quest to change your username!
Go to the Story Quest menu on the left side of the screen to access it. (Look for the Gwen icon below your profile avatar!) Open the quest menu and peek inside the New or Active story quest tabs and scroll until you find the "Make a Name for Yourself" quest!
This quest prompts you to choose a new name by tapping your existing username and adding a new one in the provided field. Not only will this quest unlock the ability to rename yourself in the future, it provides your first rename free of charge!
If you wish to rename yourself moving forward, just tap your profile picture, followed by the small feather icon beside your username. Please bear in mind that name changes after the first cost 5000 jewels.