If you have the ap you can buy a pet name tag. Then you drag icon over pet and type in your pet name.

If you buy a pet and you wish to rename it, then just follow the above to rename pet.

Pet name tags are not tradeable as of yet. If not, then if you buy one from someone, WHO CAN BE TRUSTED, then give them your pet, they name it, and give it back to you.

Do not name a pet mount as the name will not appear so it is a waste to do so.

If you are short on money then save name for the pet that will stay with you now. If you are planning to make a pet a mount, I would suggest to wait and NOT name it.