You can change your base's name by clicking on your base's HUD.
To change your nickname, simply enter your base and click on your base's current name in the top right corner of the game screen. The window to set your nickname will pop up in the center of your base.
It can sometimes take a little time for a nickname change to be reflected throughout the whole game. Please be patient. If your name still hasn't updated in 24 hours, contact KIXEYE customer support for further assistance.