To change your mate’s name, all you have to do is load a game, then click “Player stats” and edit the name put in for your mate.
Provided you have accumulated a sufficient number of experience points, you also may or may not be able to change your pups’ names. To do this, load up a game in Slough Creek and click “Pack stats”. From there, you’ll be able to modify the names of your pups after they have been born.