if there is a anivia on my team and i am the jungler

depends, if she's good you don't have to, if she's bad you probably should unless she's going to lose it.

But it is probably a good investment to give it to her if you want to just gank. Since she's a good gank partner

you should.

she is mana hungry

As a main, I generally don't ask nor really need blue buff on the 2nd/3rd spawn because of Runes/Masteries + . Then with you should have enough mana to sustain yourself and not ever really need blue in the game. Now there is some cases where Blue might be needed early, if you are facing people who are very very strong at wave clearing as well like . Since some other mages also have the same level of wave clear as Anivia with like half the mana cost as hers, or if the enemy team has baron and you really need me to clear waves of baroned minions, then yes.

I will eventually want blue buff later on in the late game, when we all have like 4 items and starting to group. Not for the Mana or even the CDR, but for the % AP that blue gives you, which is pretty huge on an Anivia with . But as for Early/Mid game, I generally leave that to the Jungler's decision to give me blue buff or not since I know blue is very important on a lot of junglers.

it depends... if she has full mana or has a first back you might not need to, but always ask.

certain mid laner may be able to get a huge advatnage if they can even use their spells to clear 2 minion waves

Depends on who you're playing too. If you're fiddlesticks you need that thing to function. Lee Sin though? Give it away to top lane if they want it.

Not really, Anivia can function pretty well without blue especially once she gets her core items but it really really helps before then.