Kytrad mentioned a forthcoming book by Ted Anthony, "Chasing the Rising Sun, The Journey of an American Song." The book is available, Simon & Schuster Canada, $32.00; not sure of American publisher but Amazon.com offers it new for $17.16 and new but with remainder marks or used from $4.49- the book already remaindered.
Anthony takes the reader on a journey through all of the speculated origins of the song. The reviewer George Fetherling, in "Books in Canada," October 2007, comments that Anthony is "expert at turning up ...fascinating but meaningless information, such as the fact that there is an album entitled "Sebastian Cabot Sings Dylan" and that the sister of Dylan's great love Suze Rotollo was Lomax's executive assistant."
The book seems to have no more information on the origin of the song than is contained in this thread, with one possible exception- "The Rising Sun Dance Hall," 'in print 1925,' mentioned by Anthony in his book.
I have only the review, not the book which may have a citation- otherwise the first appearance is in recordings by Ashley, Alexander and later Turner (see Stewie, post of 2002 above, and verses in the song "Rounder's Luck," Callahan, posted by Arkie above).
Ashley explored possible New Orleans connections, his best possibility the Rising Sun Hotel, destroyed by fire in 1822. This seems unlikely because the song appeared over a century later.
He also discusses the "Unfortunate Rake" and allies, but again, this seems very unlikely, although a number of writers on folk music have nodded in this direction.

If one likes the minutiae of a song search, he may enjoy reading this book. I may pick up one of the remaindered copies.