I love the name Ava! I think it's so classy and feminine, despite its popularity. Ava Elizabeth is a lovely combination (though I prefer the spelling Elisabeth...a little less common and softer imo). Ava Grace is nice, too. Some other suggestions:
Ava Caroline
Ava Celeste
Ava Christine
Ava Camille
Ava Danielle
Ava Felice/Felise
Ava Gabrielle
Ava Isabelle
Ava Jasmine
Ava Josephine
Ava Juliet
Ava Jacqueline
Ava Katherine
Ava Lillian
Ava Lauren
Ava Michelle
Ava Madeleine
Ava Marie
Ava Margaret
Ava Natalie/Nathalie
Ava Nicole
Ava Rose
Ava Roselyn
Ava Rachel
Ava Reese
Ava Therese