In a way, Link already has a pet and her name is Epona.
I don't know what pet he'd have. Probably a dog, to help him rescue the princess. Maybe a cat to help him relax afterwards. Either way, I bet it's a girl named Zelda.
I liked the pet chick (baby cuckoo) that link had in the Oracle of Seasons manga. It was a great wee story with those two and I have ever since thought that a Cuckoo would be the best pet