Changing my Game Center ID
I am assuming you mean changing your nickname. If so, you must go into the ME section of GameCenter, tap Account (where it says your email), hit View Account, then Edit Account Info. Scroll down to Nickname to change your name. However, I believe that the names may still be linked to the Apple ID and will therefore appear on your son's iPod. If that is the case, he may have to follow the above steps as well.