In order to rename your pet (Bombay Cat, Husky Dog, Alpaca and Peacock), click on Move button (the rake and move icon in the bottom menu), then hover over the Move Tool until you see the Name Pet Tool (purple paws pet tag icon).
Click on it, then click on your pet (or your pet’s area) to rename it, then save the new name.
Make sure you don’t change your pet’s names too often or you’ll confuse them. Kidding…