I feel like I need to run someone who can push objectives. When running jungle with lee etc, I'm at the mercy of my teammates actually taking objectives and pounding on a turret as tank lee/eve or maokai doesn't accomplish much.
Opinion time.. others may disagree with me.
A jungler has no place split pushing. Which means at no point will you be looking at taking towers solo and that lowers the effectiveness of someone like Yi. If I see a jungler doing that, that's like a green light to take control of the other half of the map. You're also giving up baron control late game if you do that.
Astrong objective jungler is someone that can gank effectively and potentially solo or duo dragon early. Not surprisingly it's the more popular junglers that can do stuff like this:
Pantheon
Kha'zix
Elise
Pantheon is my favorite. Great map control with the ult, has good AD for helping to push down those towers early when you get a successful gank and he's a strong duelist. With proper lane pressure you can extend your control into the opponent's jungle.
You're going to get tower control by influencing your laners with your ganks and your map control, not by split pushing or taking them yourself.
Again, all of this is just things that I've found success with. I know there are guys that swear by tank junglers and wreak havoc on the enemy with it.
edit:
Bonus jungler time. I've recently been enjoying Fiddlesticks. He has a lot of strengths as a solo queue jungler. Good ganks, great dragon and baron control and good dueling potential.