"stew made with whatever's available," 1904, hobo slang, probably from a proper name. The golf sense of "extra stroke after a poor shot" (1949) is sometimes said to be from the name of a Canadian golfer in the 1920s whose friends gave him an extra shot in gratitude for driving them over rough roads to their weekly foursome at St. Lambert Country Club near Montreal. The name is from Gael. Maolagan, O.Ir. Maelecan, a double dim. of mael "bald," hence "the little bald (or shaven) one," probably often a reference to a monk or disciple.
http://www.etymonline.com/index.php?term=mulligan
A "Mulligan" is a second chance in golf, permission granted by the other players to re-take a flubbed shot, especially the first shot of the game. "Taking a Mulligan" is, strictly speaking, forbidden under the rules of the game, but since the term has been in constant use since at least 1949 we can assume that such charity on the links is not uncommon. Unfortunately, no one knows where "Mulligan" in this sense came from, although we can assume that it probably sprang from the proper name of a particular person, perhaps an especially inept golfer.
As a consolation prize (and, with luck, to head next week's golf question off at the pass), I can tell you the origins of two other common golf terms. "Birdie," meaning a score of one stroke under par on a given hole ("par" being the standard set for a first-class player) comes from the 19th century U.S. slang term "bird," meaning anything excellent. The fact that such a beautiful shot "flies like a bird" probably contributed to the adoption of the term by golfers.
A "bogey" for a given hole is the score expected of a good amateur golfer, and is often, but not always, the same as par for the hole. The term was invented in the late 19th century by a certain Major Charles Wellman, who noted that the declared "standard score" of the course he was playing in England amounted to a virtual "bogey man" -- an imaginary ideal golfer -- against whom he was forced to compete. The Major later decided that such a formidable opponent must certainly be an officer, and christened him "Colonel Bogey," a term still heard today.
http://www.word-detective.com/070401.html#mulligan