The trick is, you don't have to follow the meta, but you have to understand why the meta is meta. The meta is some kind of "global optimization" on all the factors of a game. You can decide to focus more on certain factors at the price of some others, but you have to really understand the trade-offs you're making.
Something off-meta can have a lot of very powerful strengths, but there is a reason it's off-meta, so it must also have some glaring weaknesses elsewhere. As a quick example, Morgana/Lux botlane is very strong, BUT you basically can't take towers unless you wipe out the enemy team.
So, as long as you understand well your off-meta pick, why it's off-meta, what are these weaknesses and you can play around them, you can pick off-meta and have great successes.
EDIT: I've read other answers in the thread, and I want to emphasize. Don't fool yourself, if something isn't meta and you think it's super strong and the next OP thing, 9 times out of 10, you'll be wrong. If something is NOT meta, there IS a reason. As I said above, this reason doesn't necessarily mean you can't find success at all, if you understand it properly, but the meta is not just sheeple blindly following the Flavor of the Month. It's there for a reason.