It is shown in the comic that Lady Luck is Brenda Banks, the beautiful blonde socialite heiress, daughter of Bruce Banks, a millonaire mine owner.
As Lady Luck, Brenda's costume was essentially a green dress and a large green hat. In some issues she wore a gree veil in place of a mask, but otherwise she went completely maskless.
Despite this, somehow no one (not even her father or friends) could tell that Lady Luck was Brenda Banks as long as she was in costume!!!!