As you probably know, a variety of roles fit into the jungler position. You can play tanks and bruisers, and Riot's doing their best to fit mages and marksmen into that role too. So pick whatever role will fit your team.
However, from experience, the best junglers are the ones that snowball their lanes and win the game. I've seen plenty of junglers win games by hitting what seems like every lane while still counter jungling and thus stopping the other jungler from doing so. So the best junglers:
-Clear quickly (for faster ganks)
-Gank well (cc, damage, gap-closers)
-Have good sustain (to back less and thus outlevel the other jungler)
This is why Lee Sin
is popular. He has high damage and some aoe, he has 2 gap-closers and a slow, he has a shield and life steal, and best of all he's manaless. Rengar
is similar, he just has a harder time ganking pre-6. Champions with the three characteristics I mentioned will make the best junglers. Those with two of the three will be alright, and those with only one you should probably avoid.
Also, ganking well is the most important of the three categories. Keep that in mind.
Also, objectives are key. Yes, you want to hit the lanes hard. But if you can take an objective because of that, do it! You can ward as well, maybe ward in their jungle around dragon or the herald. Create map pressure and then use it to take objectives.
P.S. I'm no game theorist, all of this is my own opinion. I have, however, played since season 1, so that should count for something.