To change your Profile Name, go to the world map, open up the Game Settings (the gear icon), scroll down and choose "Edit Profile". Inside the "Edit Profile" dialog, you can enter a new Player Name. Or, if you are in a Den, you can also tap on yourself in the Members list to get info about yourself, and then choose the "Edit Profile" button from there.

Note that this used to be called "Camp Name" but we have changed it to "Player Name".