If you are level 10, you can change your name! In order to do this, tap your name at the bottom lower right corner of your game, and a menu will open up. Your name will appear at the top of this menu. Tap your name in order to change it. This opens an area where you can choose to edit your name. 
The first name change is free, and each one after that will cost 1000 diamonds. Your full name (the first and last name together) has to be unique - i.e. no one else can have the same name as you.