Aina 3

For people who might be confused about the literal meaning, I don't think the name is meant to be perceived as "a child loving vegetables". I thought it would be more like "Love and Vegetables", meaning you want your daughter to turn out to be a nice cook. That's just what I think about, with the posted kanji on the site.

Not super-common. I've only ever met 1 Aina and her kanji was 愛名, so it was love and name. All depends which kanji are chosen what it winds up meaning. Multiple possibilities.