"I want a pet because it's cute" isn't a problem with the game that desperately needs to be solved. "I want a pet to fight at my side" isn't either, 'cause we already have mechaknights and we all know how well those turned out.
On the other hand, here's an actual problem: When I go through clockworks tunnels style levels, I'll often come upon a fork in the road. Down one of these forks will be a party button or elevator, and down the others will be more monsters and uncollected loot. I want to go kill those monsters and collect that loot, but I don't know beforehand which path will take me to more loot, and which will take me towards the party button/elevator and cause me to have to backtrack through the level.
This is where the pet comes in. Let's say it's a pet mewkat or wolver puppy or something. It accompanies me in a purely cosmetic non-combat fashion through the level, wandering along by my side. This is where the useful part comes in: It will always tend towards directions where there are unkilled monsters or uncollected loot, and preferentially away from the elevator or party button unless the loot is already collected.
Thus with the aid of my pet, I will never again have to backtrack through levels to make sure I've cleared them out completely.
You, sir, are a genius.
I never really thought of pets this way...
A way of changing which path you want your pet to point out first would be awesome too, thus making pets helpfull for both rushers and looters.
Maybe changing that could involve some kind of training;
gosh... So. Many. Possibilities...