So here's how to counter Anivia:
Since Origen demolished TSM, I expect to see a large influx on "Anivia OP/Annoying" when people start to play her. So here is how you counter her:
Counterpick
Pick someone who outscales her. Anivia likes a slower game, so pick someone who can play to that. Who outscales Anivia? Karthus. The answer is always Karthus. Orianna and Azir are also good. Pick someone who won't let her reach her late game. Fizz, LeBlanc, Zed, pretty much any assassin really. Scaling lane bullies are also good choices. Pick someone who doesn't care about her slows. This means champions with lots of gapcloser or can ignore cc, champions like Irelia, Yi, or Vi. This is more for non-midlaners. Don't pick champions who have to run at her. If you pick someone who needs a Sivir to normally do their job, you're going to have a bad time.
Counterbuild Pretty straight foward. Build MR and Tenacity. Aegis is a good choice due to it's AoE armor. Hexdrinker is good since she has single target burst. Merc Treads give MR and Tenacity.
Counterplay: Deny her blue. Anivia is way better when she has blue, even when she has RoA and Tear. Don't let her have blue. Get in her face. Don't let her have an easy lane. She scales very well, and she's rather happy when she can sit back and just free farm. Gank her. She's immobile and dependent on gold, so 1 or 2 good ganks can really set her back. CC her. She's all squish, her ult is channeled, immobile, everything about her makes her vulnerable to cc. Dodge her Q. This is the most dangerous part of her kit, as there is a slow, stun, and a good amount of damage on this skill. It also allows double damage on the E. However, it is also pretty slow. Remember, dodge this left or right of it, as it has a long range and you probably won't be able to outrun it.
Hopefully, this helps with the Anivia spike I'm predicting.