I'm gonna cheat and pick one ADC, one mage, one support, and one tank, because I can't really pick one of those roles over another as my favorite.
For the mage, it's gotta be Veigar. Sometimes the random isn't so kind and puts you against a team without a high-AP caster, but he'll still hit like a truck. I just find him very fun to play overall.
For an ADC, I probably prefer Cait over the rest. And not necessarily because of her large AA range, but more for her Q. It's pretty rewarding to catch somebody with the tip of it, and just see it chunk away their health.
For the support role, Karma. She'll be played a bit more like a mage, I guess, but the AP boosts her supporting abilities. And she also has to focus on getting a lot of CDR as well, to be most useful. Some of the things she can pull off through her obscenely-low-cooldown shield and her sustain, she's a bit of a trollface to play. And I find that highly entertaining.
For a tank, I love Amumu. No real good reason, I just find him fun to play.