The name "Dionna" may be an anglicization of the name of the Greek goddess "Dione". She was an Oceanid (an ocean nymph) and, by some accounts, the mother of the goddess of beauty, Aphrodite.
The name may also be a form of the Greek name "Dionaea", which was another name for the goddess Aphrodite, derived from the name of her mother, Dione. "Dionaea" was also a name applied to objects or creatures sacred to the goddess Aphrodite.
As for what "Dione" means in Greek... You'd have to ask Pavlos that.