What does the M mean in the browning BAR serial?

No special reason known. It was given the letter M as the product code when it was introduced in 1968, until 1976 when the letter "M" was replaced by # "137" as the new product code. Occasionally the older Browning guns were given product codes that represented the gun that it was used on such as the letter "S" for Sweet 16's, but a Standard weight 16 ga A5 was the letter "R". The letter M has no special significance other than to identify the model in the serial code indexes.