It costs 20 Gems to change your Vegas World name. When you change your name, all references to your old name will be replaced with the new one, so your friends will still be your friends, though they may no longer know who you are. Any gifts you have sent using your old name will now display your new name.
To change your name:
Select the Menu Button in the upper-left of the screen.
Select Settings from the list that appears (look for the Hammer icon.)
Select "Change Name" and enter your new name selection. If the new name is not available, the game will say so. The 20 Gems will only be charged once your name has been successfully changed.