Margo, quite often I have found Minnie to be short for Wilhemina. That said, she was listed as Minnie even in the 1870 census.
Perhaps at some you point you will find a formal record that will answer the question of her first name.
I'm curious, do you have any idea why they would have gone to Alabama?
Please, allow me to Google that for you...since apparently Google doesn't work on your computer.