I had this same idea and did some research. Here are some of my findings of pets that have attacks strong against a particular family and are of a family who takes less damage (though this of course isn't foolproof as many pets have several attacks from a different family).
Note this is strictly about damage strengths/advantages, other utility/healing not included because on most families the 'counters' are extremely limited
Aquatic: Jade Owl (Magic with a good bit of flying damage) though it isn't extremely easy to get, Enchanted Broom has a little flying damage as an alternative. I believe uncommon is the only quality for either (pre-battle stones)
Beast: Haven't found one, not sure there is a flyer with mechanical damage.
Critter: Flayer Youngling (humanoid with good beast damage). Curious Wolvar pup has one attack at level 10 as an alternative
Dragonkin: Scourged Whelpling (Undead with one CD and one channeled humanoid damage ability)
Elemental: Any Snail (Not a solid counter, as though it is a critter, its one Aquatic ability is a level 20 with a CD)
Flying: As you mentioned, the Sprite Darter Hatchling is a glaringly strong counter
Humanoid: Maggots or Spiders (Both are beasts with an undead 'life leech' like ability. Not particular strong though)
Magic: Mechanical Pandaren Dragonlilng (mechanical with one spammable dragonkin attack), think it is only uncommon
Mechanical: Most any elemental, as elementals are both strong offensively and defensively against Mechanicals. I prefer the Phoenix hatchling so far.
Undead: Most Toads (Aquatic with good critter damage)
Hope that helps.



