A sweet, nostalgic nosegay kind of name, Posy has been long fashionable in England, a country of gardeners, but this pretty bouquet-of-flowers name is still rarely heard here-- though it could be seen as a more unusual possible alternative to Rosy or Josie.
Posy is a little sister name in the Hunger Games series. Before that, Posy Fossil was a character in Noel Streatfeild's Ballet Shoes.
Posy (or Posey) could also work as a nickname for a range of other names, from Penelope to Sophia--it started as a pet form of Josephine. Other P-beginning flower names you may want to consider along with Posey: Poppy, Petal, Primrose.