Astrid - This keeps your "A" theme going nicely. It's a GP for me, and I can't decide whether I like it or not.
Emerald - I agree with Meghan, I can't see someone with Emerald as a first name though it could be okay as a middle.
Evelina - I love this name. I could definitely see Athena, Aviana, and Evelina as sisters.
Honor - I feel like this is too stiff and quite a bit to live up to for a girl's first name (or even a middle). Sorry.
Lilia/Lillia - I think this is really pretty. I actually know a Lillya, but I prefer your spellings. However, I'm not sure it would fit as a first with your daughters' names.
Penelope - This is another one that I'm not sure about my feelings for, but I do like it to an extent. Though it has a bit of a different style, I think it's cute in a sib-set with your daughters' names.
Primrose - I do like this, although I feel like it's so closely associated with The Hunger Games that it's struggling to stand on its own as a name. I'm not sure this one fits with Athena and Aviana, but that's just my opinion. It would make a lovely middle, though.
Rosalie - Very cute, and I can see it going well with your daughters' names.
Brooks - This is one I'm on the fence about, though I do think it's handsome. I guess I could see Brooks fitting with Athena and Aviana, but maybe not as well as some others.
Daxton - I'm not a fan, but Dax is a cute nickname for it.
Finley - Cute, but I feel like it's (for lack of a better word) "casual" compared to your daughters' names, if that makes any sense.
Flectcher - I prefer the spelling Fletcher, which I do really like (and it has a nice meaning).
Leo - I have negative connotations with this name, so I'm a bit biased.
Levi - This is the one that I think goes best with your daughters' names.
River - I really love this name. I see River the same way as Brooks in that it fits okay with your daughters' names but not as well as a couple others.
Roman - This is probably up with Levi on my favorites from your boys list to go with your girls.