in the ggClassic Epik vs CLG game, CLG ran a "kite" comp.
They basically ran fast move speed champs that had very good poke and would kite epik until they were low enough to collapse on. It worked so well it was scary.
I have also seen true damage comps, with vayne, olaf, cho. It is kind of one trick but the focus target is guaranteed to die.
For a college tournament, I had a few comps pre-planned based on certain team synergy tricks, not really high elo or anything more theorycraft on beastly mechanics. We ended up using a couple:
Super stun gank heavy - Sion, taric, ashe, veigar, udyr
suppress heavy - malz, ww, scarner, taric, trist
Generally speaking? No, probably not.
Mostly because AP characters value levels over items in most cases, and in bot lane you'd be having a support sucking up XP. Overall, this would mean that the enemy's AP carry in mid is becoming stronger than the AP character in bottom lane, while the allied AD carry in mid isn't becoming notably stronger than the enemy AD carry at bot. AD carries scale with items, and not really with levels. Thus, the AD carry wouldn't really gain anything from being in mid lane.
The majority of AD carries also couldn't lane against the majority of AP carries. While AD carries would start off strong, with decent pokes, once the AP carry gets to level 6 and has a Doran's Ring or two it is pretty much game over unless they've fed. There is no way an AD carry can keep up with the raw control and damage of an AP mid. Regardless of who the mid is.
The only exceptions to the above rule would be Corki, Ezreal, and possibly Urgot. Corki actually does scale with levels, as does Ezreal to an extent. Urgot is just innately tanky with super long ranged harass. ALL of those characters would have mana issues though, and would have to rush a mana regen item to compensate for that. And Corki and Ezreal would still have trouble with high burst/high CC champions such as Cassio/Annie/Ahri/Karthus/Just About Every Popular AP Mid Right Now.
Specifically speaking? Sure!
There will always be specific situations where you could counter certain champion picks by using more elaborate strategies. This is where a lot of the original "meta breaking" strategies come from. Such as running kill lanes in bot, double AP carries, double AD carries, or even double jungle. It can work in certain situations, but it won't work on a regular basis. And will rarely if ever work in solo or duo ques.
Best Answer
There's three things you want in a split push champion.
The first is the ability to take down towers quickly. You only have so much time before the enemy team figures out you're there, and you need to make the most of it. Similarly, you want to be able to clear out minion waves quickly, though this is a smaller issue as a lot more champions can do this easily.
The second is dueling capability. Often times the enemy team might send only one or two champions to stop your push while the rest try to continue what they're doing. Certain champions excel at winning 1v1's and even some 1v2's, and if you're playing one of those, you can just kill them and continue pushing.
The third quality is escape ability. Sooner or later the entire enemy team is going to come after you, and at that point it'd be a great idea to try and get out alive.
A very good (and very cheap) split pusher is Master Yi. His abilities include a massive AD boost, a massive AS boost, an incredible movement boost, and even resistance to slows. This is literally everything a split pusher could ever ask for, as you'll be able to murder towers and lone enemy champions, and he's nearly impossible to catch.
However, there's three other things you should do to increase the chances of succeeding with your split push.
The first is to choose your spots well. If the enemy team is currently teamfighting your allies in the bottom lane, by all means, go take out a couple top lane towers. If three of them are dead and you have vision on the other two, again, push to your hearts desire. If they're all live and none of them are visible? That's a bad time to push.
The second is to ward well. There's this nifty 75 gold item known as a sight ward, that lets you see a small area of the map. Plant these things at choke points around the area you plan on pushing, and you'll know when the enemy team is coming for you. This makes escaping much easier, and is likely to save your life several times through a match. (They're also very important for normal laning and teamfighting, but that's another issue.)
The third and final thing is the teleport summoner spell. If you've got a huge minion wave pushing top but you're currently back at base, teleport there and you'll get the tower down before they know what hit them. On the contrary, if you've just taken down an inhibitor and you know they're coming for you (thanks to those wards you placed earlier), you can use it like a much faster recall to get out of there... possibly to another lane you can push down.
On the flip side, keep in mind that teleport is also an amazing way to stop a split push as well. If you see three enemy champions with teleport, you might be better off just trying to teamfight. Similarly, if they have abilities that can effectively emulate a teleport (Twisted Fate, Nocturne, or Pantheon's ultimates), you need to be a lot more careful, as they can appear at any time and try to kill you.
One last thing, although split pushing is a valid and extremely effective tactic, keep in mind that there are times where it's just better to teamfight. If the enemy team's main push would kill your nexus before your backdoor can destroy theirs, you need to go back and help out your team. This can also apply to fights around Baron, as winning that buff is often enough to seal the game. Then again, in the EU vs NA all star game, a split push during a baron fight took out everything in NA's base short of the Nexus itself, so use your own judgement.
Good luck, and happy split pushing!