I like the name in general. It does remind me of the Beatles song, "Dear
Prudence," which I know primarily from the film 'Across the Universe'. I like the character of
Prudence in the film and the song is nice, so it's not a negative association.
I'm not so keen on Prudy for a nickname, but
Prue or
Pru are nice. They do remind me of 'The Hunger Games' a bit - possibly because of 'Prim' being used as a nickname for
Primrose (Katniss' younger sister). ("Prim and proper" is similar to "prudence"? Sort of vaguely anyway.) Again, it's not a negative association for me.