I usually play a civ called "random" whose name changes right after the game starts.
They must be my favorite civ, as well. Although I would use this option more often if it was possible to exclude some civs from the poll. (I dislike playing the same civ two times in a row, for example, and I rarely feel like playing a warmonger civ.)
If I had to choose, though:
Vanilla: Iroquois - I don't know why exactly but I can recall I just loved to play them when the game first came out
DLC: Babylon and Korea: I love the science game, and also being ahead in science means easier access to wonders.
Also Spain. It's a gamble but I love the NW bonus. Mount Sinai + the right Pantheon + Spain = 20 faith per turn! I would have more fun with the conquistadors though if the AI ever bothered to settle the new world on terra maps. I stopped playing Spain on terra maps because I would just snowball from renaissance onward with a whole new continent all for myself.
GnK: Maya - I find their UA extremely cool, and they have one of the best UB in the game (for me at least: again, more science and practically a guaranteed religion)
BNW: I can see myself playing Shoshone a lot because of the Pathfinder. It will be my new Spain. Also Poland because I'm Polish and their UA is extremely adaptable. And probably Indonesia because I love the "colonization" aspect of their UA and the randomness of their Kris Swordsman.