When you set up your iPhone you will be assigned a random nickname in Game Center. And you wouldn’t want to be recognized by a random name in the gaming world or else how you can brag about your conquests ;)
I used to ignore changing the game center nickname but finally decided to change it at last.
You can change iPhone Game Center nickname easily in following steps.
You will be prompted with 3 options. Tap on View Account. You might need to enter your iTunes password here.