A New York minute is an instant. Or as Johnny Carson once said, it's the interval between a Manhattan traffic light turning green and the guy behind you honking his horn.
It appears to have originated in Texas around 1967. It is a reference to the frenzied and hectic pace of New Yorkers' lives. A New Yorker does in an instant what a Texan would take a minute to do.
I'll have that ready for you in a New York Minute.