Of American-English origin, the colloquial phrase off the cuff means without preparation, extempore, on the spur of the moment.

The image is of notes jotted down on a shirt-cuff; for example, in the following from The Film Daily (New York) of Friday 25^th March 1928, Jack Cohn (1889-1956), co-founder of, and producer at, Columbia Pictures, was described as writing on his shirt-cuff during the 14^th semi-annual golf tournament held at the Rockville Country Club the previous day:

Somebody said Jack Cohn had “stymied” and Jack wrote it on his cuff as a good title for a future Columbia release.

The earliest instances that I have found of off the cuff in its current sense indicate that the phrase originally referred to scenario improvising during the silent-film era—and not, as often claimed, to the notes from which a public speaker can improvise.

The first occurrence is from Writer Will Come into Own in Sound Film Era, Cohn Says, by Alfred Cohn¹, published in The Film Daily (New York) of Sunday 7^th October 1928:

With the coming of the “talkie” script, prepared with meticulous care, with every word of dialogue carefully weighed and every sound tested, the director will find himself in his right place, an interpreter just as is the stage director. The supermen of the megaphone will no longer “shoots [sic] ’em off the cuff” after ostentatiously destroying the scenario. The intelligent director will work more closely with the writer because he must look to the writer for the life that is to be breathed into the moving shadows.

¹ Alfred A. Cohn (1880-1951) was an American author, journalist, newspaper editor, Police Commissioner and screenwriter; he wrote in particular the screenplay of The Jazz Singer (1927), the first feature-length movie with synchronised dialogue, which marked the end of the silent-film era.

The second-earliest instance that I have found of off the cuff in its current sense is from Making Whoopee: Ziegfeld² Would Have Found Things Far Different Had He Started Turning Out His First Film in the Early Days of the Cinema—How to Start to Become a Film Actor, published in the St. Louis Post-Dispatch (St. Louis, Missouri) of Sunday 2^nd November 1930—this article is about “Florenz Ziegfeld’s first try at the movie game, “Whoopee”, with Eddie Cantor³ as the star”:

Colin Chase⁴, who is in “Renegades” at the Fox Theater, was in the films at the time when Ziegfeld was getting his first toe hold on Broadway […] and he recalls some of the troubles of that period.
“[…] Scenarios were unknown: the director either made his pictures entirely ‘off the cuff’ or he had a brief sort of synopsis to go by, and made up his own scenes as he went along.”

² Florenz Edward Ziegfeld Jr. (1867-1932), American Broadway impresario
³ Eddie Cantor (Edward Israel Itzkowitz – 1892-1964), American comedian, dancer, singer, actor and songwriter
⁴ Colin Chase (1886-1937), American actor

In his column Hollywood Sights & Sounds, published in the State Times Advocate (Baton Rouge, Louisiana) of Tuesday 24^th November 1931, Robbin Coons also used the phrase when writing on the evolution of the professional relations between film directors and screenwriters during the transition from silent movies to talkies:

The latest formula for insuring movie hits—an air-tight script before a camera is set up, with director and writer working together as a team—has been successful in a number of instances, relieving to some extent the worries of executives who feel keenly the need of money-makers.
Yet, one picture that has its makers doing joyous nip-ups has been shot “off the cuff,” in old quickie fashion—that is, the director shot each day what the writer had finished the day before.

However, the earliest instance that I have found of off the cuff has a different signification; Oscar Odd ‘O. O.’ McIntyre (1884-1938) defined to live off the cuff as meaning to mooch in his column New York Day by Day, published in The Richmond Palladium and Sun-Telegram (Richmond, Indiana) of Monday 8^th February 1926—that day, O. O. McIntyre was writing from, and about, Miami, Florida:

There are very few of those dapper and gaily-habited young men who enhance the cosmopolitan air of Manhattan, down here. Instead you see the butter-faced country boys with apple cheeks and the rolling gait of those who follow the south end of a plow in the broiling sun. It is evident they have rushed here hugging dreams of escaping the monotony of farm drudgery. Many will return to an earlier environment reconciled. Thousands are living “off the cuff,” or as we say up north, “mooching.” And, of course, there are the thousands who have shucked off a mediocre humdrum for the opulence of wealth acquired almost over night [sic].

The American novelist Jacquin ‘Jack’ Lait (1883-1954) used to get off the cuff with in the sense of to get even with in the following dialogue from chapter 27 of The Broadway Melody, published in The Herald (Dayton, Ohio) of Friday 19^th April 1929:

[Queenie] “Did I ever ask you for anything? Did I ever tell you I gave a whoop for you? Did I?”
[Warriner] “N-no—not exactly. But in the sweet language of Broadway you hinted at it.”
“I ain’t heard any sweet language of Broadway. An’ just exactly when an’ how did I hint that you meant a thing in my young life?”
“You went out with me—you took—”
“Yeh. That was my mistake. I took. Well, if you’ll let go my hands a second I’ll give you back the bracelet. I’ll send you back this gown as soon as I can take it off. That grand you sent me—I ain’t got it now. But I’ll pay you back if it’s a nickel at a time till I get off the cuff with you—so that’ll leave us fifty-fifty.”
Warriner looked into her face searchingly to see if she could really mean it. He saw no sign that she didn’t.
“No, that won’t even us up,” he answered.