I've been playing her a lot recently (some big success and some big failures). Personally I find she doesn't yet have a lane opponent she "can't beat" in essence, she can safely stay at turret, making dives virtually impossible thanks to passive, stun and wall. Then once you do hit 6, you can waveclear so hard with a short R and a long Q. I can push a lane under turret, and attack the enemy who is under the turret from a safe distance, denying them farm SO much. As long as you watch out for enemy jungler. I often run Boots of swiftness on her because he speed is SO horrible, even with ionion boots and alacrity, I often move slower than most other champs around me.
Most of my failures have come from bad decision making (greed) and dodgy mechanics (I need a new mouse, I often cast spells in the wrong direction or off target :/ ) but the biggest cost of playing Anivia is her need for a team play. On her own, she can wreck almost anyone late game 1v1, as long as you have good timing and accuracy. But in a team fight, it's hard to focus one person, as soon as you weaken them into death range, they back off and a full hp champ comes on you. If you don't have a decent peel (like thresh or maokai) or if your team chases blindly leaving your ulti and passive debuffs, chasing with her is a nightmare. By the time you catch up, half your team is dead. Anivia is primarily defensive, and is very bad at chasing. You need a good peel to defend against a whole team, baron and super minions (and they still won't take your turret), but you need an assassin or global ulti to take them out then they retreat. If your tanks are chasing (nasus or malph, etc), they can get kited and killed before you catch up (if not running swiftness). I often have to flash just to be in the game :/
I say all this because I see her as VERY strong, and very applicable in multiple scenarios. If she got buffs she would dominate. I think slightly higher ms and shorter passive at higher levels sounds great. But you can actually control her mana quite well, and if it was reduced, it's make using her SO MUCH EASIER. If i didn't run 2 mana regen items, then I'd have something like 950-1000 AP or something defensive (like GA which can be a bitch on Anivia lol) and with 2 100% ratio skills, that's over 2K damage with two hits. Not including ulti, and not taking into account the low CD on E which means another 1K damage before the enemy can get around your crystalze or recover from your stun. I tried running only 1 mana item, but I was struggling late game to keep going. I could optimise in a number of ways, but I'm experimenting right now (my match history shows how bad some of my experiments have gone) but it'd only take a small reduction in mana costs to allow me to significantly alter my build and build paths, so yeah, ms sounds good (low just not THAT low) and scaling passive.
Edit! No, there is one thing she needs desperately above anything else. Fix that dam Auto attack. I swear, sometimes it does "less" damage than what her base stats suggest, on top of the slow cast time and slow travel time. I understand if they want to make her farming harder, but VERY low on hit damage, with Very slow travel speed and VERY slow cast, makes it too ridiculous. Change one of these things, you don't need 3 gates on JUST aa's, they are always gonna be weak, so you can at least make them land reliably. She will never compete against Ori or Lux, but at least let her last hit a caster minion, cause even those are often too much.
or give her AA's a bonus like her E, so they do more on hit damage if you've frosted the target, it'll still cost a lot of mana early, but at least you stand a chance at a pre-six wave clear, a strong opponent can have more than double your farm in less than 8 minutes, because of her other weaknesses, don't make the little farm that she can get so hard to hit too :(
