Why can't I change my Game Center nickname it never changes?

Howdy Jason5555,

It sounds like you are trying to change your Nickname in GameCenter but it doesn't seem to save. I would sign out of GameCenter:

Sign in with a different Apple ID on your iPhone, iPad, or iPod touch

Game Center

1. Tap Settings > Game Center. If you see the option to sign in, you can change your Apple ID now.
2. Tap your Apple ID, then tap Sign Out.

Then quit the Settings app:

iOS: Force an app to closeDouble-click the Home button.

Swipe left or right until you have located the app you wish to close.

Swipe the app up to close it.

When you have done that restart the device and test the issue again:

iOS: Turning off and on (restarting) and resetting

Thank you for using Apple Support Communities.

Regards,

Sterling