as in I spoke off the cuff or an off-the-cuff kind of response.

closed as general reference by tchrist, user19148, Matt E. Эллен, user2683, Barrie England Oct 15 '12 at 14:15

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http://www.etymonline.com/index.php?allowed_in_frame=0&search=off+the+cuff&searchmode=none says

Off the cuff "extemporaneously" is 1938 American English colloquial, suggesting an actor or speaker reading from notes jotted on his shirt sleeves rather than learned lines.

It's fairly common for students today to write notes on their hands or wrists. I guess in the 1930s it was common for people to write notes on their shirt cuffs -- well, if not common to actual do, at least an idea that many were familiar with. So if you were going to give a speech but hadn't actually prepared something and written it up, you might jot some hasty notes to yourself on your shirt cuffs. Then you were speaking "off the cuff" rather than from a prepared text.