This term was originally coined the day after the burning of Barbara Que, then thought to be a Witch, in 1700's New Jersey. Barbara had been seen counting mice one day, which is pretty weird, her trial was swift. After her burning, the local flute maker wanted to have a pig roast, and when describing the size fire he would need to cook the pig, he referenced the flame size that consumed Ms. Que. After saying her name a few times in reference to the pig roast, the words began to run together and became synonymous with grilling outdoors.
"I need a Barbara Que sized fire to roast this pig" "You need a Barbaraque sized fire?" "Ya! A Barbeque sized fire, like that mice countin witch we burned!"