Instructions
Turn on your Xbox 360 console and a controller and navigate to the "My Xbox" menu.
Choose your Gamertag, then select the "Change Gamertag" option.
Select the "Enter New Gamertag" option if you already have a new Gamertag in mind or choose "Get Suggestions" if you would like to see some automatically generated options.
Press "Yes, use this gamertag" once you have selected or entered in a new Gamertag to apply your modification. If you don't have enough Microsoft Points (800 are needed), you will be prompted to add more.