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?
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Please, allow me to Google that for you...since apparently Google doesn't work on your computer.