My favorite Gym in terms of type specialty is the Fuschia gym...we need more poison-specialists that aren't members of evil organizations. >_>
In terms of the puzzle, my favorite has to be the ice in Sootopolis Gym. It was fairly easy, but still made you think, and because I'm the type of person that intentionally challenges all the gym's disciples for the experience, I had to do the ice puzzle twice each game. Once to fight all the trainers in the basement, and one to actually reach Wallace/Juan.