Every once in a while, champions that were thought to be best in other lanes suddenly become popular as supports. Annie, Zyra, and Morgana were all originally intended to be played as mid laners, for example, and Fiddlesticks, Elise, Gangplank, and many other champions have also moonlighted as supports throughout the years. Even Amumu, Galio, and Kennen have seen competitive play in bot lane in the LCS.
There are plenty of reasons to stray from the beaten path when it comes to your support picks. Primarily, if none of your opponents know how to play against what you've picked, you're already at an advantage. Even an experienced bot lane will have to think twice when faced with a new type of opponent, and depending on the champion you choose, that advantage could carry over to later game, with the entire enemy team unsure of what to do in the face of an unusual team composition.
That said, there are some champions that definitely do not work as supports, so you have to think carefully about which champions can work as supports and which can't. So what makes a champion viable as a support?
It used to be that in order to be viable as a support, a champion must be able to do its job with virtually no gold. However, with the advent of ward restrictions and the new support items, that's changed a little bit. In the current meta, you should look for a support that meets the following qualifications:
Must be able to make a significant impact without high offensive combat stats (AD or AP).
Your team needs you to buy a Sightstone, and you need to buy yourself a gold generation item and boots. It might be a while before you get around to buying yourself some combat stats, and even when you do, you won't have as much as you would in another role with two more open item slots, so you want to make sure that your champion has an ability or two that will have significant impact no matter what your items are. Morgana has a 2-3 second snare and a CC-blocking spell shield. Thresh has incredible engage and disengage. You need to choose a champion that has abilities that have just as much impact.
If you don't buy a Sightstone, you're either warding less than you need to or spending too much on wards. The ward restrictions were a great way of encouraging the other roles to buy more wards, but make no mistake, warding is still a focal part of any support's job.
It's possible to do well as a support without a gold generation item, but only if you're able win your lane hard and get a lot of assist gold for yourself. You don't want to put yourself in a position where losing or going even in lane makes you behind in gold for the rest of the game. Supports don't need gold as much as the other roles, but it still makes a big difference. A gold discrepancy between you and your enemy support could mean that the other team has an Aegis or a Mikael's and you don't.
All of that said, don't be afraid to buy combat stats after your Sightstone and gold generation item if that's what's best for the situation. If you're playing a champion that benefits from but doesn't need combat stats, then a Rabadon's or Void Staff might be the best choice for your next item. Sometimes your team could use that extra damage just as much as it could use a Mikael's.
Should be able to utilize one of the gold generation items effectively.
As I stated above, it's a big risk to rely on assists or towers as your sole source of income. If you aren't making money off of your gold generation item and your enemy support is, you're at a big disadvantage.
Must be able to create pressure during laning phase.
Helping your marksman through laning phase is one of your most important jobs as support. A fed or underfarmed marksman can make or break the game, so you can't choose a champion that won't create pressure in lane. If the champion that you're thinking of supporting as doesn't make an impact until level 6, you should reconsider playing that champion.
Must be more or equally impactful as other supports that you can play.
You shouldn't just be asking yourself if it's possible to support as a certain champion, but also whether it's worth doing. Just like in game, you should always be thinking about opportunity cost during champion select. Theoretically, yes, you could run Tryndamere as a support, building Relic Shield, maxing Mocking Shout to win trades in your lane, and using your ult to facilitate tower dives, but unless you have more or as much impact on him as you would on another champion, then it's just not worth choosing over that champion. Only if you can find a way to make that Mocking Shout just as game-changing as a Leona ult or Thresh hook would Tryndamere be worth taking as a support.
You must know how to play to your champion's strengths and weaknesses.
This is really the case for any role, but it warrants emphasis because your champion's strengths and weaknesses as a support may be different than they would be in another role. If you're playing a champion that specializes in engaging, don't hang around on the back line. If you're there to peel for your marksman, make sure you're never far from them. If you're there to lock down a single target, make sure you choose the right one.