You can change your pet's name at any time. 

Tap the "^" button in the bottom left corner and select Design Portfolio. Enter the Character Hub and tap on Pets. Choose the pet (cat or dog) you want to rename, tap the Edit button, enter the new nickname using the keyboard that appears, and then tap "OK" to save the change.

Please note: Currently, you can only change the name for your cat and dog unlocked through the main storyline.