From all the evidence I've been able to locate, I'd have to believe that his name was Huddie Ledbetter. However, that is based not on any defining fact, but on the preponderance of evidence. That includes John and Alan Lomax, Charles Wolfe, Kip Lornell, and several other "lesser" researchers. Just as his birthplace (Texas or Louisiana) has been in dispute, so has his name. To resolve these questions, one weighs the arguments and decides on the basis of who said what, and how much each of those persons might have known.
I have actually seen the census report that states "Hudy" (note this spelling!) was born January, 1888.
No ACTUAL middle name has ever been found.