This column by Adam Freedman discusses the phrase:
The Oxford English Dictionary lists the first printed reference to a hung jury in Edwin Bryant’s What I Saw in California (1848-49) in which he states: “The jury . . . were what is called ‘hung’; they could not agree . . .”
Bryant’s phrasing obviously suggests that the phrase was already in common use by the late 1840’s. ... The earliest use of the term in a law report appears in an 1821 case, Evans v. McKinsey. ... it appears that the term developed somewhere in the south during the early 19th Century.
Linguistically, the phrase seems to derive from the sense of “hung” to mean caught, suspended or delayed (“I got hung up at the office”).
The Hung Jury: The American Jury's Insights and Contemporary Understanding [PDF] references the same cases, with a few more details (see footnote, pg 1).
Summary
The American cut your stick, to die, comes from the British cut your stick, to depart, which dates back to at least 1813.
Americanisms
Uncle Tom's Cabin by American author Harriet Beecher Stowe was published in 1852 so let's check an almost contemporary dictionary.
Maximilian Schele de Vere's Americanisms; the English of the New World (1872) says on page 594:
Irish origin?
The OED has the British English sense of departing from 1825 but with no etymology.
The Wordsworth Dictionary of Phrase and Fable (2001) based on the original by Ebenezer Cobham Brewer (1810 – 1897) offers an Irish origin:
I must cut my stick--i.e. leave. The Irish usually cut a shillelah bfore they start on an expedition, Punch gives the following witty derivation:--
"Pilgrims on leaving the Holy Land used to cut a palm-stick, to prove tey had really been to the Holy Sepulchre. So brother Francis would say to brother Paul, ' Where is brother Benedict ?' 'Oh (says Paul), he has cut his stick ! ' — i.e. he is on his way home."
OED antedating
The OED can be antedated in Grose's Classical Dictionary of the Vulgar Tongue (1823, page 107 -- but not in the 1788 or 1796 editions) which simply says:
CUT ONE'S STICK. To be off. Cant.
Further antedatings
Further antedatings can be found in Othello-travestie: In Three Acts, with Burlesque Notes in the Manner of the Most Celebrated Commentators and Other Curious Appendices by John Poole (and William Shakespeare) (page 8):
Roderigo.
Why not cut your stick ? (b)
Page 29, just before Cassio leaves:
Cassio.
I'll cut my stick.
And in the extensive footnotes:
Given the subtitle of the book -- "with Burlesque Notes in the Manner of the Most Celebrated Commentators" -- I don't think we can trust these notes to be real, and therefore the 1597 to be fictional, but they are at least 1813 examples.
Best Answer
The bare expression beat them off with a stick is quite old, and started out as a literal statement of how to get rid of pests:
The earliest example I can find of a figurative extension of this concept to the idea of warding off romantic suitors who are so numerous as to be a nuisance is in the story A Man's Man by Ian Hay, originally published in Blackwood's Magazine in 1909:
I'm not sure whether the description of this phrase as "inartistic" suggests that it was Hay's own coinage or an already-established-but-clumsy cliche. However, it appears again a few years later in a romance novel:
P.G. Wodehouse also used it in 1923, but to describe a different human annoyance:
By mid-century it seems to have been fairly well-established in its current formulation:
The earliest figurative usages are all by British authors, so I suspect this usage is British in origin. It seems to have been popularized to some extent in the context of the American entertainment industry later in the twentieth century, however, and usage of the verb beat rather than keep may partially be an American contribution.