I loved when we could tame the Alterac Valley wolves who ran as fast as the rocket mounts seem to. It was a blast seeing them chase any moutned player down in seconds.
My first hunter was a Dwarf at launch, I remember taming Sian Rotam the white lion which is a horde only skin and the only available back then. I was lucky to get a horde player to help, he ended up taming the the other spawned cat in that quest and I got my white lion. I also remeber how happy I was taming my first raptor.
I'd say what I miss the most is when some random skin had a special attribute. There was Deathflayer the 1.6 speed scorpion, the wolf with resists, and a bunch of other cool pets. Eyes of the Beast is sorely missed as well. It was a blast in battlegrounds while defending to use that ability and take people on with your pet. Blizzard lost major points doing away with EOTB.
My most noteworthy recent tame was the white serpent in Caldera I think it was, that was a interesting tame. I also managed to tame Kirix the green spider, Banthalos, Quilen. I also tammed Garwal the Worgen and was fun using him while he lasted.