Since
Molly is a nick name for
Mary, it has the same meaning as
Mary. Just as
Cathleen,
Katherine and
Kate all have the same meaning.
Molly has also been used as a nn for
Margaret.
But even if it was a nn to begin with, I think it can stand alone as a formal name. Just don't look for a special meaning for
Molly. The origin is of course English and Irish.
"In Dublin's fair city, where girls are so pretty, I first set my eyes on sweet
Molly Malone."
"You may give them your love but not your thoughts. For they have their own thoughts.
You may house their bodies but not their souls,
For their souls dwell in the house of tomorrow, which you cannot visit, not even in your dreams."
(On Children by Kahlil Gibran)