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Pet rock Never leave your mountain halls without it !

That piece of humor aside, I'd imagine dwarves being a truly industrial sort of buggers that don't usually take a liking to small, fuzzy, adorably cute things.
Their pets should serve some specific purpouse, and not just loiter around and poop on their backyards.
If it cannot be ridden, eaten, or provide some other use it's just a freeloader.

Deep underground, where fires may cause a suffocation hazard, and fuel is a rare and expensive commodity.
Fungus (althoug not strictly a pet as much as a decorative plant) and fishtanks can provide illumination (I imagine a white, nearly blind fish wich uses a natural "lantern" to lure its prey close to its mouth or to scare off predators)

Insects might be used as "watchdogs" or even beasts of burden if they are large enough, and again illumination is a possibility with insects as much as fish.
Just because its of a monsterous size doesn't automaticly make it a "monster".
Although the thought is slightly alien and possibly "off-fantasy", I can envision domesticated giant beetles pulling ore carts and dwarven riders. Or even be used in the mining process itself if it where to poop out the ores/gems and digest the rock (think purple worm MM p.214 or similar) or just plain tunnel building.

However we might not want to step into Drow teretorry here (I'm not very experienced with any AD&D setting yet so I'm not sure whats kosher)

Other, more mundane species of worm are great to make useful soil out of nasty sewage wich might otherwse be a enviromental hazard (clogging tunnels, causing poisonous gasses, or pollute freshwater sources)
Certain types of cavernous lizards could perhaps be used as "pidgeons" to carry messages between the halls or as "canaries" for mining purpouses.

Bats, goats and other mountainous "livestock" like ponies, yak's and llama's could be found in settlements closer to the surface. The bats are a particularly nice idea that I used for quite some time (not to take credit from Great_Zamboni).
I imagine them a common ingredient in the dwarven culinary arts. And young dwarves often go "fishing" for them with net, sling and spear.
But deeper under the mountains all of those animals would be outside their natural habitat, and getting food for them might be more of a hassle than its worth, especially if kept in flocks or herds.

If these suggestions seem to "flintestones" for you, perhaps you'd like a more "warhammer" flavour, consider mechanical constructs, or perhaps stone golems in unorthodox shapes.

Hope this was useful !
I'm a big fan of dwarves myself, and quite curious what others might come up with

Skunk would work really well, with the tunnels limited breeze (or constant breeze, depending on the tunnel) their stink spray would either linger for a long while quite effectively as a 'wall' of poison gas or pass away in the next round acting more like an 'attack' of poison gas. A giant skunk would of course be better as they are able to be mounts as well at that point.

Hmm...Is it just me being inexperienced with AD&D settings vs general fantasy settings here ?
All these suggestions sofar strike me more as something a halfling/hobbit type would agree with, as they are mostly woodland or field dwelling creatures.
Not saying they are bad suggestions though...They just strike me as odd is all.

I'd guess Dwarves away from home would keep the same type of animals humans do...As dull as that may sound.

My favorite pet as a dwarf seconded as my mount...it was a giant lizard...I had a Dwarven Ranger TWS...high levels in riding. Picture a giant lizard barreling towards you on the ceiling of the cave, with a Dwarf on its back, battle axe in one hand and hammer in the other. Scary sight.