I love Ava, and I wish it wasn't as popular as it is. I only know one little Ava IRL, but I know of others at my son's preschool--though none in his class--and more than a few of my online friends have Avas as well. It's definitely not the right choice if you want your daughter to be the only one with that name in her class.
I like Aviva, and Avia is new (to me) and interesting. I think you could use Ava as a nn for either, but she would still have a more distinctive fn to fall back on in school.
If you like the Eva family of names but are worried about popularity, Eve could be a safe bet. While Eva, Ava, and Evelyn are climbing and climbling toward the top of the charts, Eve is holding steady in the 500-600 range. It actually decreased in popularity in the most recent year.