"jury-rigged", or "jerry-rigged"

Merriam-Webster's Collegiate Dictionary, Eleventh Edition (2003) says that jerry-rigged goes back only to 1959. It speculates that the term is an amalgam of jury-rigged (dating to 1788) and jerry-built (dating to 1869). The jury in jury-rigged doesn't involve a panel of one's peers, however; it means "makeshift" and appears in the Middle English jory saile meaning "makeshift sail." The term jury-rig thus means (according to the Eleventh Collegiate):

to erect, construct, or arrange in a makeshift fashion.

As for jerry-built, Farmer & Henley, Slang and Its Analogues (1893) offers this discussion:

Jerry-builder, subs. (common).— A rascally speculating builder. Jerry-built, adj., = run up in the worst materials. [The use of the term arose in Liverpool circa 1830.]

From the Facts on File Encyclopedia of Word and Phrase Origins (2000):

jerry-built. The cheap, flimsy constructs of a Mr. Jerry of the Jerry Bros. of Liverpool may have inspired the word jerry-built. Jerry-built could also be connected with the trembling crumbling walls of Jericho; the prophet Jeremiah, because he foretold decay; the word jelly, symbolizing the instability of such structures; or the Gypsy word gerry, for "excrement." Still another theory suggests a corruption of jerry-mast, a name sailors and ship builders gave to makeshift wooden masts midway through the last [19th] century. Jerry-masts or rigs derive their name from the the French jour, "day," indicating their temporary nature.

So whereas jury-rigged suggests "improvised in an emergency," jerry-built signifies "very shoddily constructed."

The much later jerry-rigged splits the difference, according to the Eleventh Collegiate, but perhaps tends a bit closer to jerry-built than to jury-rigged:

organized or constructed in a crude or improvised manner.

UPDATE: The Dubious Jerry Brothers [8/11/14]

In a column headed "Notes on Books, &c." in Notes and Queries (January 26, 1901), an uncredited reviewer discusses the newly released Oxford English Dictionary, Vol. IV, Green—Gyzzern and Vol. V, Invalid—Jew, under the editorship of James A. H. Murray. After noting "Few parts of the 'Dictionary' are more interesting than that dealing with the letter J, the growth of which is exceedingly curious," the reviewer observes in passing:

No satisfactory origin for jerry-built, jerry-builder, &c. has been found, one put forward in the press deriving it from a Liverpool firm of builders not standing investigation.

A followup item by Murray headed "'JERRY-BUILD' : 'JERRY-BUILT.'," in Notes and Queries (April 20, 1901) details the investigation that the earlier reviewer alluded to:

I may add that, after seeing the original letter to this effect [namely, to the effect that the jerry in jerry-built referred to "Jerry Brothers, builders and contractors," which supposedly was "a Liverpool firm in the early part of last century"] printed in Truth in January 1884, I wrote to its author asking for the evidence on which the statement was made. In his reply, now lying before me, the writer admitted that no evidence was producible ; he added that he was under the impression of having heard this explanation of jerry-builder from the English master at the school which he attended, but he had subsequently searched for authority without finding any ; and Sir James Picton, our great Liverpool authority, who had been consulted, had never heard of it. He therefore could not maintain the reliability of the story, and frankly withdrew it. In preparing the articles on the Jerry words in the 'New English Dictionary' (section published 1 January last) we made further investigation, with the help of correspondents in Liverpool, and ascertained that no trace of any such name as Jerry in connexion with the building trade could be found.

The late-nineteenth-century investigation by Murray and his correspondents in Liverpool strongly suggests that the attribution of jerry-built to the slipshod work of the oddly untraceable Jerry Brothers is apocryphal.

(Credit for the discovery of this Notes and Queries content goes to EL&U stalwart Peter Shor; see his comment below.)

Although the Jerry Brothers seem to have vaporized under scrutiny, there is a Liverpool connection to early use of the term. In a book published by John Murray twenty years after the Notes and Queries episode, Ernest Weekley, An Etymological Dictionary of Modern English (1921) remarks:

I conjecture that jerry-built may be for jury-built, the naut. jury, as in jury-mast, being used for all sorts of makeshifts and inferior objects, e.g. jury-leg, wooden leg, jury-rigged, jury meal, etc. Its early connection with Liverpool, where jerry-building is recorded in a local paper for 1861, makes naut. origin likely.

As for the theory that the word is of Gypsy origin (mentioned in the Facts on File Encyclopedia of Word and Phrase Origins excerpt above), we have this item from Barrere & Leland, Dictionary of Slang, Jargon & Cant (1897):

Jerry. This word is common among the lower classes of the great cities of England in such phrases as jerry-go-nimble, diahrrœa; jerry-shop, an unlicensed public-house with a back door entrance, and jerry-builder, a cheap and inferior builder who runs up those miserable, showy-looking tenements, neither air-proof nor water-proof. Jerry seems derivable from the gypsy jerr or jir (i.e.,jeer), the rectum, whence its application to diarrhœa, a back door, and all that is contemptible. From the same root we have the Gaelic jerie, pronounced jarey, behind ; the French derriere.

I took a quick stab at an Ngram comparing jury-rig, jury-rigged, jerry-rig, and jerry-rigged; I'm well aware that Ngrams are not definitive, but they do give a good overview. "Jury-rigged" is by far the most common of the four constructions. "Jerry-rig" and "jerry-rigged" don't seem to have come into use until after World War II; I speculate that that may have been due to "Jerry" as a slang term for "German".

According to the American Heritage dictionary, "jury-rig" comes from "jury-mast" (a temporary replacement), which probably came from Old French ajurie, "to help".

I don't think we can say that either "jury-" or "jerry-" is correct or incorrect - they're clearly both in common use - but "jury-rigged" is both older and more commonly used.

Edit: addressing a secondary question within the question - "rigging a jury" (more commonly called "jury tampering") has nothing to do with a temporary repair; that's a different meaning of both "jury" and "rig".