Vanity is generally used in a negative context, so I don't understand why someone would call their child this. Vanity is defined as "the excessive belief in one's own abilities or attractiveness to others" or "something that is vain, empty or valueless". Ouch. Maybe they thought the word was pretty and didn't fully understand what it meant? It's also a type of bathroom or bedroom furniture, making the name all the stranger of a choice. I would just use Verity or Amity.
I would just use Verity or Amity.