I've heard a lot of stories of how REALLY GOOD players played only one champion before learning new ones, even getting to Challenger playing only one.
I was thinking about doing this with Ahri, but what if she gets banned? I could pick up another champion for that event, but that would no longer be maining only one champion.
Sorry if the answer is obvious; I'm new around league and I don't know my options in such a situation.
Picking up a second champion is not "maining" a second champion. I find it's best to know your main extremely well, but dick around with other champs just for fun. That way, when Ahri gets banned, you have a second mage-type champ you can play.
Focusing on one will allow you to play better, that's for sure. I suggest maining one champion (just make sure they're not a constant ban) and be competent with at least one or two others.
I don't really like the idea that 'maining' a champion means that you play him/her any chance you get. I think of it more as a champion you've played more than anything else and are more likely than not going to pick if you can. There are easy counters in this game that, even if you're highly skilled with your champ, a goof with basic mechanics can give you a hard time with. There are also times when your 'main' just doesn't fit the team comp properly so unless you're going to carry hard early it's not going to be a good pick.
Playing primarily one Champion is great, it gets you to a high level of mastery with them. Just make sure you're competent with a slew of others. There WILL be a time when Ahri is banned. There WILL be a time when you have to play something other than mid. The last thing you want to do is get shoehorned into jungling and have no idea what you're doing or who to do it with. You'll be staring down a BIG loss.
The champs that people did this with were not popular champs, and were almost never picked/banned
Main means to pick whenever you get the chance. Obviously there are lots of reasons you may not get that option: banned, picked by enemy before your turn, teammate ahead of you in Draft also mains the same role (this is by far the most likely unless you're maining a Perma-ban). For these cases just try to be competent enough in at least 2 other roles and be reasonably good on at least 1 other champion in your main role.
E.g. I main caitlyn adc, but I'm also an excellent tristana adc. I'm also a good ahri mid, and a good Morgana support. So I have 3 roles I can play well, and 2 champions I can play very well in my best role. That said, I'll grab caitlyn whenever I get the chance (although I'm only level 24 on Draft normals, so I actually grab mid whenever I get the chance because everyone wants it so it's hard to practice it). I'm also one to be diverse and cooperative so I play jungle when asked to as well and am willing to try top even though I fail at it, but I guess that's irrelevant to your strategy :p
I main Garen and I pick him almost every chance I get, but I also play Nasus, Leona, Sona, Master Yi, Nautilus, Viktor, Syndra, Nunu, and Sion. I am noticeably better on Garen compared to the other champions, but I am able to play each of these champions without feeding (usually).
Basically your main or mains are champions you are trying to get good enough to carry with under almost any situation, where as your secondary champions are champs you only try to be competent with.
Or what everyone in bronze does: troll until someone dodges. 
Technically maining a specific makes it incredibly easy to raise your ranking, especially if someone never sees that champion. If that champion you main is banned, nothing you can do and you just pick something else. If its made unable to play that champion, you have to play something else. You will still be an Ahri main, but if you play something else doesn't mean you aren't anymore. To technically just be an Ahri main, you'll need to have her as your most played champion, so just go with that go and you're set.
If you like mid lane, chances are that you will only get to choose that lane maybe every 2 in 5 times you play anyway. "Maining a champ" is thus not exactly possible for the majority of your games. You really need to be comfortable with about 10-15 champs
I just try to play different roles while maining a role. Sometimes your main champ can be counter picked or be bad for your team.
Yeah, you should never only play one champion. Always have at least three champs for each role that you can at least play competently if your main gets banned/if you get stuck in a different role. 3 toplanes, 3 mids, 3 supports, 3 ADCs, 3 junglers. Yeah people can get to really high elo by playing one champ, but it is rare that they play that champ exclusively. The more champs you can play well, the better your chances of being able to play against the other team through knowing counter picks and match ups. Also, what would you do if they nerfed Ahri into the ground or if they disabled her while they changed something (Hypothetically). You would be left without ANY options at all and would have to either flop around like a fish out of water and lose all the time, or have to stop playing league until they fixed her. The best players don't have to rely on only being really good at one champ, because they spent the time early to learn a LOT of champs well.
