As soon as your dogs have had a puppy and the big balloon comes down, select the basket and choose 'Rename'. Name the dog. After you have done that and become a little bit of a friend with the puppy, select 'Start' then go to 'Personal Info'. Look at your relationships. You will find the puppy's name, but the picture will show it grown up. If you do not think the name you chose suits how the pet looks, then go back and rename it.