"bottom of a sleeve," mid-14c., cuffe "hand covering, mitten, glove," perhaps from Medieval Latin cuffia, cuphia "head covering," which is of uncertain origin, perhaps ultimately from Greek.
Sense of "band around the sleeve" is first attested 1520s; sense of "turned-up hem of trousers" is by 1896. Meaning "a fetter for the wrist" is from 1660s. Adverbial phrase off the cuff "extemporaneously" is attested by 1938, American English colloquial, suggesting an actor or speaker reading from notes jotted on his shirt sleeves rather than reciting learned lines. Cuff-links (also cufflinks) is from 1887.
"to strike with or as with the open hand," 1520s, of unknown origin, perhaps from Swedish kuffa "to thrust, push." Related: Cuffed; cuffing. As a noun, "a blow with the open hand," from 1560s.