Maybe someone better than me should try this...
I had an idea today. Since the support item changes really opened up the board for supports, I thought, .. what about Evelynn?
Hold on, .. hear me out.
First of all, her passive gives her a good amount of health back for free, .. as long as you get to general safety with 1HP, her passive will heal you up to somewhere around 50% or more.
250 − 590 (based on level) (+ 250% AP) health
Her Q, Hate Spike, is pretty good harass. You can trigger 3 uses of Spellthief's Edge and Electrocute or, the better option imho, Phase Rush. Phase Rush allows her to disengage while harassing, and chase when she, and her carry, have pressure to secure a kill.
Her W, Allure, is a 1 - 2 second charm. It is pretty difficult to land straight-up in laning phase, ... because the enemy can see you. It does work fairly well for zoning though. Even while ahead, people generally want to avoid getting charmed. Once you hit 6, it becomes easier. The downside it does let the enemy know roughly where you are.
Her E is not great for this in my opinion. I think it makes her commit a little too much, but when you CAN commit, it does help her to gap close and the bonus 30% movement speed can work in her favor as well. Again, this is something you can get away with a lot easier while in stealth or if you can get someone to face check a bush.
Speaking of being in stealth, .. Being in stealth actually gives your carry a lot stronger presence while ACTUALLY on their own, and allows them to get some solo experience. As long as you don't show on a ward or gank a lane, the enemy has to be at least a little concerned that you're there. This doesn't mean you just abandon them though. You do need to BE present enough to make the enemy laners think that you are likely around when you really aren't.
And of course, her R is an execute, and can always be used as a "get out of certain death ... almost free" card.
So I did try this in a Normal. I was solo queued, but I asked the team if they would be okay with it. Surprisingly they were. I do believe it went alright, but I was hard counter picked. 6 of the 11 games their support played TODAY was on Galio, including this game. If I had to guess, I would say they're a one-trick, and they played like one. Their Flash > W or E > W's were pretty much perfect, and he just stomped me early. I don't mean to blame our loss completely on him, but I do think a better match up might have turned out better.
However, once I hit 6 I was able to start putting some real pressure onto other lanes, especially their jungle, which I think is where she would really excel. Just make sure to swap to the Oracle's Lens so you can be sure you aren't spotted when you invade and you can work with your jungler to deny farm from the enemy jungle.
So again, someone better than me should try this. I honestly think it could be a good pocket / cheese pick in the right match up. I'm definitely not suggesting that it is a great pick, or that it should even become meta. There are WAAAY better options, but it could be fun.