Minnie was wildly popular at the turn of the last century—it was the fifth or sixth most popular name throughout the 1880s—but is completely obscure today. Blame Mickey's girlfriend. Regardless, it's possible that the up and coming trend toward old-fashioned nickname-names—think Maisie, Mamie, Millie—may give Minnie (all on its own, not as a short form of anything) a new moment in the sun. Minnie Driver (born Amelia) has given it some modern celeb cred.
There have been characters named Minnie in Dickens's Little Dorrit and Eugene O'Neill's Mourning Becomes Electra. Cab Calloway's "Minnie the Moocher" is a jazz classic.
Minnie can really be short for just about any M name (and some non-M names as well, such as Wilhelmina), including Minerva, Mary, and Margaret.