sign-off n.

early 13c., signen, "to make the sign of the cross," from Old English segnian and Old French signier "to make a sign (to someone); to mark," both of them from Latin signare "to set a mark upon, mark out, designate; mark with a stamp; distinguish, adorn;" figuratively "to point out, signify, indicate," from signum "identifying mark, sign" (see sign (n.)).

The sense of "to mark, stamp" is attested from mid-14c.; that of "affix one's name or signature to" is from late 15c. The meaning "communicate by hand signs, make known by significant motion" is recorded from 1700.

Transitive sense in baseball, "engage (a player) by the signing of an agreement" is by 1889. To sign out (transitive) "secure the release of (someone or something) by signing" is attested by 1963, of library books. The intransitive sense of "record one's departure" is recorded by 1951. Related: Signed; signing.

by c. 1200 as an emphatic form of Old English of (see of), employed in the adverbial use of that word. The prepositional meaning "away from" and the adjectival sense of "farther" were not firmly fixed in this variant until 17c., but once they were they left the original of with the transferred and weakened senses of the word. Meaning "not working" is from 1861.

Off the cuff "extemporaneously, without preparation" (1938) is from the notion of speaking from notes written in haste on one's shirt cuffs. In reference to clothing, off the rack (adj.) "not tailored, not made to individual requirements, ready-made" is by 1963, on the notion of buying it from the rack of a clothing store; off the record "not to be publicly disclosed" is from 1933; off the wall "crazy" is 1968, probably from the notion of a lunatic "bouncing off the walls" or else in reference to carom shots in squash, handball, etc.