Chiffy
"Etymologicon Magnum, or Universal Etymological Dictionary" by Walter Whiter (1800) makes the claim that "chiffy", as used in the term "in a chiffy" derives from the Anglo-Saxon word "Caf".
"A Dictionary of the Anglo-Saxon Language" by Joseph Bosworth (1832)
confirms the meaning of "Caf" as "quick, sharp, nimble, swift".
Jiffin
This is my oldest source yet, this time for "jiffin".
"The Fall of British Tyranny: or American Liberty Triumphant" by John Leacock
This was published in MDCCLXXVI, which by my reckoning is 1776.
Please to walk aft, brother soldiers, that's the fittest berth for you, the Kidnapper's in the state room, he'll hoist his sheet-anchor presently, he'll be up in a jiffin --- as soon as he has made fast his end of his small rope athwart Jenny Bluegarter and Kate Common's stern ports."
Jiff/Jiffy
In 1791, Edward Nairne of Sandwich, Kent published "Poems, Miscellaneous and Humorous, with Explanatory Notes and Observations" in which the following lines appear:
At dinner-time, and bus'ness slack,
I stept to Joe's, and got a snack
A pot of mildchee, and a whiff,
And off again in half a jiff !§
The author's explanatory notes, below, are expansive and delightful:
§ Jiff or jiffy, a jocular expression, and means a short space of
time. Innumerable are the expressions (particularly amongst sailors)
to shew what expedition may be, or is intended to be made, in the
doing of any act ; the progress of these is curious. I perfectly recol-
lect, when a school-boy, an expression of this kind — ' Before you
can say Jack Robinson' — was very common. After the intervention
of various others, that of — ' As soon as you can say peas' — came
into vogue ; but some persons, who were not over precipitate, very
properly qualified it by adding — ' and boil them.' Next, the ele-
gant expression of doing any thing ' In a pig's whisper' came into
fashion! (What particular period of time this contains, I am at a loss
to determine, having never yet had the pleasure of hearing these melodious animals exhibit in this way ! — I have frequently, and with
admiration, observed them make transitions from one note to another,
and which usually has a most charming effect.) — The ingenuity of
modern times has, I believe, brought this business to its ne plus ultra,
its greatest perfection ! and people can now, according to their own
declarations, do things ' In less than no time ' This beats Joshua's
making the sun stand still -, for that only protracted daylight, and
puzzled the clocksmiths ! but this has all the advantages of time,
without the inconvenience of waiting for it.
In Old Norse, Ratatoskr means "drill-tooth" or "bore-tooth". It is the name of a mythical creature, a squirrel that runs up and down the tree of life called Yggdrasil, acting as a messenger between two arch enemies: the great eagle and the terrestrial dragon.
According to Albert Sturtevant, "[as] far as the element Rata- is concerned, Bugge's hypothesis has no valid foundation in view of the fact that the [Old Norse] word Rata (gen. form of Rati*) is used in Háv[amál] to signify the instrument which Odin employed for boring his way through the rocks in quest of the poet's mead [...]" and that "Rati* must then be considered a native [Old Norse] word meaning "The Borer, Gnawer"
Wikipedia
One can therefore assume that the word rat stems from a description of its behaviour.
Best Answer
Summary: Cagey may have originated in boxing in 1890s America, and used in a more familiar way the following decade.
Caged animals' behaviour
Owen Lattimore's 1990 China memoirs suggests:
This is plausible; a caged animal can be wary, cautious and careful.
Dictionary review
After checking 33 dictionary definitions listed by OneLook, most say origin unknown and/or c. 1910 America, including Merriam-Webster, but claims "First Known Use: circa 1893".
However, Dictionary.com Unabridged ("Based on the Random House Dictionary") defines it as:
And also gives an earlier date range and explicitly gives the root as cage:
Additionally, a 1950 Slang To-day and Yesterday by Eric Partridge says:
Cagey boxers in 1890s America
Here's some possible early uses.
From The Weekly Herald, August 21, 1892, in a preview of a night of boxing entitled "Scene And Personnel Of the Great Fights at New Orleans, September 5, 6, and 7.", written by Mike Donovan. The last fight of the night:
Later that year, cagey again shows up in a boxing context. The Morning Herald of November 1, 1892 has report on "Choynski The Victor: He Knocks Godfrey Out in Fifteen Rounds" at Coney Island on 31st October:
Next, the variant cagy shows up in another boxing report, published in both the Warsaw Daily Times and Aurora Daily Express (and probably the Atlanta Constitution) of March 9, 1893:
1900s American newspapers
The following decade we can find more familiar uses of the term. I've noted which of these are pay-per-view as I've not paid to verify the OCR is correct.
Pay-Per-View - The Sun - Apr 6, 1900
A story on a "Magical Wine Cask" in the Lawrence Daily World of Oct 2, 1901 uses cagey twice to mean careful:
Pay-Per-View - Detroit Free Press - Jul 6, 1902:
Pay-Per-View - Detroit Free Press - Aug 28, 1909:
The Evening Sentinel of Aug 19, 1910, in an article about former president Theodore Roosevelt titled "The Colonel Is Saying Nothing. Refuses To Discuss Breach Between Himself and Taft":
1900s American serial dime novels
Cagey is used in the modern sense in the New York nickel weekly Secret Service: Old and Young King Brady, Detectives.
The Bradys and Dr. Hop Low; or, The Deepest Mott Street Mystery of January 4, 1907 by A New-York City Detective:
The next year: The Bradys and Little Chin-Chin; or, Exposing an Opium Gang of January 31, 1908:
Footnote: 19th century Scottish cheerful
There's an earlier Scottish term cagey which may be unrelated, and means cheerful or joyful.
From the addenda of Two ancient Scottish poems: the Gaberlunzie-man, and Christ's Kirk on the Green by James V (King of Scotland), John Callander of 1782:
There's also another note about Cadgily which links it to cadgers who carry goods for sale in cages, also called creels, "who use to sing, in order to beguile the tediousness of the way. From primitive ca, cad, cap, anything made for containing, possibly from Gaelic cadhla.
This Scottish word is used in Blackwood's Edinburgh magazine: Volume 9 of 1821:
And in 1821's Annals of the parish: or, the chronicle of Dalmailing by John Galt:
Meikle is a variant of mickle: chiefly Scottish meaning great, much.
And more from John Galt, 1834's The literary life, and miscellanies: in three volumes: Volume 3, has a cheerful meaning of cagy:
And again in the same volume:
Footnote continued: early 20th century American sexual desire
A snippet from 1917's Dialect notes, Volume 4 by the American Dialect Society says:
And a snippet from their 1928 Dialect notes, Volume 6 says:
This has roots from the Scottish cagey, joyful, as Michael Montgomery 2006's From Ulster to America: the Scotch-Irish heritage of American English explains: