Aquamarine is the birthstone for March, and what a pretty springtime color it is. I've seen a lot of unusual names being considered for middle names these days, such as Reverie, Fable and Wren. Even as first names. And I cannot think of a better middle name for a March baby (especially with blue eyes) than Aquamarine, if unusual names grace your list.
In 2010 there were 6 baby girls named Aqua, but none named Aquamarine. As far as middle names, I have no way to know. (Although, my guess is that generic middle names are still the most predominant, such as Marie and Rose.)
FYI, Aquamarine was also a movie abouts Mermaids. Aquamarine is a variant of beryl, and Beryl was also used as a given name, but is now considered vintage and dated.