Anodyne
I like this name for a girl because: A) it's the name of my favorite all-time album and song (by Uncle Tupelo); B) it means something that soothes. For short, we would call her Ana, Ann or perhaps Annie.
Here's what Merriam-Webster has to say about Anodyne:
noun
Latin anodynos, from Greek andynos, from a- "not" + odyne "pain"; probably akin to Old English etan "to eat"