Don't usually play Anivia, but who hasn't wanted to have a go at playing as the ice bird? At first I thought she was really squishy and too many dependent, but I managed to pick up a few kills. What I was aiming for was to boost:
1) Mana
2) Ability Power
3) Gold
That way Anivia could seem viable and 'keep up' in champion battles.

Courtesy

Clarity:
This will help out early game. Because Anivia is quite mana dependent, and you want as much farming/harassing as possible early on, Clarity is a good option. Also, because her ulti Glacial Storm has an 'on-off switch' activation and a really short cooldown, this is what you want to use more often for both farming and finishing off enemy champions. Though it may not be as useful late game, it can prolong the casting time of Anivia's ulti.

Ghost:
This is to make up any lack of speed early game. Also, it'll help in chasing enemies (as I have found out, Anivia cannot catch up to some in a simple chase). Furthermore, it'll be a lifesaver if you 'misplace' a Crystallize.

Heal:
Can be a viable option, especially if you know you're playing a serious game, or if you continually dive towers. But if not, her egg-form passive should be enough for you to escape/survive, for the egg's hp is full bar. The cooldown for the passive is 4 minutes.

Revive:
No.