The relevance of Annie's bag
 

What's in it that makes Annie carry it to battle?

I think she just thinks it looks cute and brings it with her.

a memento from her family? has some food in it for travelling?

Her Mommy gave her a bag full of snacks and band-aids :)

The more relevant question would be, what is it in little girls that makes them carry bags with them?

An endless supply of teddybears; She can't transform Tibbers back.

It's where she carries her items, like Health Potions.

The sandwiches her mummy made her, together with her bottle of strawberry milk. She likes to share them with Amumu after a hard day's work of pyromania and embalming.

Mom told her to pack a lunch with her in case she gets hungry idk
Place to keep Tibbers safe?

Um... there's one problem about the whole Tibbers thing. Annie still has him in her hand.
And to Arcane Azmadi, Amumu drinks the choco milk. Tibbers must take the strawberry.