From Dave Wilton:

"This golfing term for a free swing dates to 1949. The origin is unknown.

It seems likely that there was once a really poor golfer named Mulligan whose partners would allow him extra strokes, but if true his identity is now lost to the ages."

http://www.wordorigins.org/

This is my go-to guy for origins of words and phrases. I don't know another good place to look except the OED but the print is too tiny. I hope tsuwm or one of his ilk will chime in.