The origin of "Dixie" is obscure, but the apparent first use of the term in print dates to Dan Emmett and his minstrel song, "Dixie's Land," 1859. The New York Herald published the song April 4, 1859, upon the occasion of a performance by Bryant's Minstrels- "In Dixie Lann whar I was born in, Arly on one frosty mawnin,...." The spelling departs from Emmett's, probably how the reporter heard it. The article discusses "Dixie's Land, another new plantation festival." (This from The Oxford English Dictionary, 1987 Supplement)

I think most southerners trace the name to the banknote and "dix," but with little reason. How widespread was the use of the note?
When our coinage was named, Jefferson in 1785 came up with disme, from the French dixième, meaning a tenth. The first pronunciation was deem, but soon became dime. (From H. L. Mencken, "The American Language")

In 1861, G. Putnam published a book referring to the South as Dixie. "Before and After the Battle, a Day and Night in Dixie," was the title. Very soon, the name spread across the country.

(The southern part of the land occupied by the Mormons, extending into Arizona, was called Dixie. The Mormons made a strong wine which they called Dixie)