Allow more fine-grained configuration of when the game should start the player with a pet – for instance the ability to say “if I am a Knight, start me with a pet; otherwise don’t start me with a pet.”

An option to have your starting pet assigned a random name. The name pool would probably come from a hardcoded list (or even a text file like engrave.txt).

Give starting pets some form of protection against zombies’ instakill bite attacks, likely by having them start wearing a hard helm or allowing the player to put one on them.

Psuedodragons that are tiny D-class monsters. They may be the starting pet of Wizards, but don’t grow up. Instead, they have some kind of telepathic/psychic/magic-enhancing passive ability that scales with their level.

Your starting cat can have kittens.

A role whose starting pet, if eaten, confers amazingly good benefits, so that the player has to make a hard choice of whether to deliberately kill and eat their pet, and suffer the penalties.