Funny, yesterday was the first time I heard that misnomer. Apparently its been around a while.
AR was Armelite (my spelling is not quite correct). There was a pretty good article by James Fallows that appeared in The Atlantic, late 70s or early 80s. It was about the development history of the M16 which was the underlying story behind its failure in the fields of Vietnam. As often happens, at the center of the issue was egos, profits, and too many cooks. But anyway, the AR-15 was the basis for what the military adopted and the AR is the company it came from. Kindof like the B in BAR. (and yes, those problems with the M16 are now long resolved. Unfortunately it cost a lot of lives before that happened).