I just figured this out this week. It's actually REALLY REALLY easy and never fails. You play as the sim who is currently the leader, then do "club > resign from club" to the sim you want to be the new leader. Your leader will quit the club completely and leadership will pass over to that sim you were talking to. Probably best to make sure those two sims are the only sims in the conversation. Then if you want your old leader to still be in the club, just not as leader, they can rejoin as a regular member. I've tried this three times and it worked perfectly every time.
What *doesn't* work is if your leader just quits the club without resigning to another sim. Don't do that. I've had clubs stuck without a leader at all that way.
