Here is one account:
The one truth about barbecue seems to be that people use what they've got. In Texas it's beef, in the Carolinas it's pork, and in Western Kentucky it's mutton. Thanks to the tariff of 1816, wool production in the then Western United States became profitable and suddenly people found themselves with a lot of sheep on their hands.
Any story of the origin of barbecue starts with a meat that is too tough and undesirable to be sold for a profit. Mutton barbecue is no different. Aging sheep who no longer produced good wool became a virtually unlimited resource, but the meat was too tough and too strong tasting to be worth anything so people turned to the tried and true methods of low and slow cooking. In the early days a whole sheep would be cooked for long hours over a low fire. A mixture of salt water would be mopped over it and it would be served up with a dipping sauce of vinegar and hot peppers and stuck between a couple slices of bread. In Kentucky this "sauce" is called a dip, specifically Mutton Dip or Vinegar Dip.
Call it the Protectionist Theory of Barbecue, plus or minus a bit of hysteresis. I've seen or heard of mutton barbecue only in Kentucky and then only parts of Kentucky, the southwest and a bit in Lexington. I wonder if they have mutton barbecue in North Africa or the Middle East. In general it is an open question why barbecue traditions have for so long been so geographically concentrated.
From The New Yorker, here is another account:
How come this is the only area where mutton is barbecued?" I asked an Owensboro merchant who had been kind enough to give me change for a nickel parking meter.
"I expect because there are so many Catholics here," he said.
I didn't want to appear ignorant. "Yeah," I said. "I suppose that'd do it."
As I was searching my mind for some connection between the Roman rite and mutton consumption, the merchant told me that the large Catholic churches in town have always staged huge picnics that feature barbecue and burgoo–burgoo, another staple of Owensboro barbecue restaurants, being a soupy stew that I, for some reason, had always associated with southern Illinois. In the early days, the church picnics apparently served barbecued goat. In fact, Owensboro might have arrived at barbecued mutton by a process of elimination, since people in the area seem willing to barbecue just about any extant mammal. In western Kentucky, barbecue restaurants normally do "custom cooking" for patrons who have the meat but not the pit, and among the animals that Posh & Pat's offers to barbecue is raccoon. The Shady Rest, one of the most distinguished barbecue joints in Owensboro, has a sign that says "If It Will Fit on the Pit, We Will Barbecue It. It is probably fortunate that the people of the area settled on barbecued mutton as the local delicacy before they had a go at beaver or polecat
In other words, they don't know either. What would Robin Hanson say?: Something like: "Food isn't about eating!"
I thank Brandon Sheridan for the pointers.