Did or didn't the "jury-" prefix (jury-rigged, jury-mast) derive from "injury?"
Solution 1:
Some background on the 'jury' in 'jury-rigged' and 'jury mast'
Merriam-Webster's Eleventh Collegiate Dictionary (2003) offers the same information about jury in the sense of (for example) jury-rigged that Peter Shor notes in a comment above:
2jury adj {M[iddle] E[nglish] jory in jory saile improvised sail} (15c) improvised for temporary use, esp. in an emergency : MAKESHIFT {a jury mast} {a jury rig}
Injury, on the other hand has this derivation, according to the Eleventh Collegiate:
{M[iddle] E[nglish] enjuren, fr. A[nglo-]F[rench] enjurer, fr. L[ate] L[atin] injuriare, fr. L[atin] injuria injury} (15c)
For its part, The Compact Edition of the Oxford English Dictionary (1971) has no entry for jory.
The only instance of jory saile from a fifteenth-century source that a Google Books search discloses appears in an entry for jury-sail in Middle English Sea Terms, issue 20 (1958) [combined snippets]:
jury-sail
1468–70. Yarmouth Court Rolls ... j velo magno et ij bonettes j meson seil j jory saile ... j virga pro le water seill.
[1592. . . . jury-mast. See Weekley s.v.]
[1616. Smith, Descr. New Eng. 50. (NED.) We had reaccommodated her a Iury mast, and the rest, to returne for Plimouth.]
"Weekley" is Ernest Weekly, An Etymological Dictionary of Modern English (1921), which offers this interesting entry for jury-mast:
jury-mast. An obscure naut[ical] witticism. Cf. synon. F. mât de fortune.
I was left all alone, and let me drive in the sea five days before I could make my jury-mast {Capt Thompson, 1592}
That the "jury-mast" in Captain Thompson's account was a makeshift is evident from the slightly longer excerpt from the original that The Mariner's Mirror, volume 54 (1968) provides:
Falling off from the carrack [sailing ship], when she was yielded, my mast went by the board. Then I was left all alone, and let me drive in the sea five days before I could make my jury-mast.
Likewise, the Captain John Smith instance refers to a makeshift mast, as evidenced by this longer extract from his Description of New England (1616):
In the end I was furnished with a Ship of 200. and another of 50 [tons]. But ere I had sayled 120 leagues, shee broke all her masts; pumping each watch 5 or 6000 strokes: onely her spret saile remayned to spoon before the wind, till we had reaccommodated her a Iury mast, and the rest, to returne for Plimouth.
Joseph Shipley, Dictionary of Word Origins (1945) [combined snippets] offers this discussion of jury in its two fundamentally different senses:
jury. This body of sworn men gets its name, from OFr. jurée, sworn, from L. jurare, jurat—, to swear, from jus, jur—, law. ... But the word jury, in such combinations as jury-mast and jury-leg, temporary (a wooden leg), is for the day?, from OFr. jornal, jurnal, journal, daily, from LL diurnal— from L. diurnus, from dies, day. Thus a journey was originally a day's march; a journeyman, a worker by the day; a journal, a daily record. Journal was first an adjective; ca. 1500 the noun (account, record, register, etc.) was dropped; by the end of the 16th c[entury] its doublet diurnal (coming into use ca. 1550, meaning the list of religious services for the day) had replaced journal as the more formal adjective for daily. Johnny cake is a folk change from journey cake.
Shipley's analysis is intriguing—and in fact it echoes a word-origin theory that goes back to Francis Grose, A Classical Dictionary of the Vulgar Tongue (1788)—but it doesn't seem to have generated a lot of support from subsequent etymologists and commentators. Robert Hendrickson, The Facts on File Encyclopedia of Word and Phrase Origins, fourth edition (2008) has this very brief entry for jury rig:
jury rig. A temporary repair made on a boat to replace a broken part, most often by using something crafted on board. Ultimately from the Latin adiutare, to help. "We jury-rigged some fishing gear."
The Wordsworth Dictionary of Phrase and Fable (2001), based on a similarly titled book by Ebenezer Brewer, who died in 1897, mentions jury-leg as a humorous extension of the use of jury in the sense of makeshift:
Jury Mast. A temporary mast, a spar used for the nonce when the mast has been carried away. Th origin of the term is unknown; it has been in use for certainly over three hundred years, and was probably a bit of sailor's wit.
"Jury" has been humorously tacked on to other nouns, giving to the word a makeshift or temporary significance, e.g. Jury-leg, a wooden leg:—
I took the leg off with my saw ... seared the stump ... and made a jury leg that he shambles about with as well as ever he did.—[Walter] Scott: The Pirate, ch. xxxiv.
But in Brewer himself, Dictionary of Phrase and Fable (1900) says nothing about the humorous sailors, instead confidently subscribing to the "mast for a day" theory of jury-mast:
Jury Mast. A corruption of joury mast—i.e., a mast for the day, a temporary mast, being a spar used for the nonce when the mast has been carried away. (French, jour, a day.)
Unfortunately (but not altogether surprisingly), neither Brewer nor anyone else who argues for joury mast as the source of jury mast (which authors have done as recently as 1980) has pointed to any instance of "joury mast" or "joury-mast" in a seventeenth-century or earlier English text.
Most instances of jury-leg that a Google Books search finds—which are not terribly numerous but go back at least to Grose's Classical Dictionary of the Vulgar Tongue (1788) and in nondictionary usage to John Davis, Jack Ariel, Or, Life on Board an Indiaman (1847) and Jam Hannay, Singleton Fontenoy, R. N. (1851)—occur in a nautical context, which tends to support Wordsworth's impression of that particular term as a jocular maritime extension from jury mast.
Where did the 'injury' theory of 'jury-' come from?
One propagator of the injury theory appears to be James Stormonth, who is cited in Charles Mackay, The Gaelic Etymology of the Languages of Western Europe: And More Especially of the English and Lowland Scotch (1877):
JURY MAST.—A mast hastily constructed by sailors in a storm, to replace a mast that has gone by the board.
A supposed corruption of injury mast.—STORMONTH.
Probably from French jour, a day; i. e. a mast for the day.—WORCESTER.
Gaelic.—Diugh (diù pronounced jiù), to-day; re, during; i. e. a mast for the day, a temporary mast.Sailors also call a wooden leg a "jury" leg, because it does temporary duty.
"Stormonth" is James Stormonth, Etymological and Pronouncing Dictionary of the English Language (1876), which has this subentry within its entry for jurist:
jury-mast, n. (a supposed corruption of injury-mast), in a ship, a temporary mast placed instead of another one lost or carried away, as in a storm.
Stormonth may have drawn his suggested etymology from The Cabinet Dictionary of the English Language (1871), which is notable for voicing less equivocal support for the derivation in question:
Jury-mast, n. {Probably for injury-mast, one carried away by the injury of weather.] A temporary mast erected in a ship, to supply the place of one carried away in a tempest or an engagement, &c.
Hostility to the 'injury' theory
William Whitney, The Century Dictionary: An Encyclopedic Lexicon of the English Language (1889), however, is far from persuaded by the injury-mast attribution:
jury-mast n. {The element jury-, found first in jury-mast and later in similar naut. compounds, jury-rudder, jury-rig, jury-rigged, and the slang term jury-leg, is usually supposed to be an abbreviation of injury; but this presupposes a form injury-mast, a highly improbable name for a new mast substituted for one which has been lost. The accent also makes an abbr. to jury- improbable. More improbable still are the etymologies which refer the word to Dan[ish] kiőre, a driving, ... or to journey (a journiere mast, i. e. a mast for the day or occasion") (Grose). It suits the conditions best to take the word as simply jury + mast, it being prob. orig. a piece of nautical humor, designating a more or less awkward mast hastily devised by the captain and carpenter consulting as a 'jury'.} Nut. a temporary mast erected on a ship, to supply the place of one that has been broken or carried away, as in a tempest or an engagement.
Walter Skeat, An Etymological Dictionary of the English Language, third edition (1898 [first edition was in 1881]) shares Whitney's view that injury-mast is an unlikely source of jury-mast, although he comes out in favor of the Danish theory over the humorous "jury + mast" theory that Whitney prefers:
JURY-MAST, a temporary mast. (Scand.?) 'Jury-mast, a yard set up instead of a mast that is broken down by a storm or shot, and fitted with sails, so as to make a poor shift to steer a ship;' Kersey, [Dictionarium Anglo-britannicum] ed. 1715. Of unknown origin. β. Doubtless a sailor's word, and presumably of Du[tch] or Scand[inavian] origin. A probable source is Dan[ish] kiűre, a driving, from kiőre, to drive; common in compounds, as in kiűre-hest, a draught-horse, kiőrevei, a carriage-way. Cf. Norw[egian] kyőre, a drive, a journey without a stoppage; Swed[ish] kűra, Icel[andic] keyra, to drive. In this view, a jury-mast is one by help of which a vessel drives along. ¶ The supposition that it is short for injury-mast is most unlikely, owing to the difference in accent.
The fact that Skeat doesn't address the humorous "jury + mast" theory is regrettable, especially since that theory seems to have become, perhaps by default, one of the leading etymological explanations today.
But the obscure nautical joke explanation arose at a time when the earliest known jury- term was jury-mast from either 1616 or 1592. That jury- has subsequently been discovered more than a century earlier in the form jory saile complicates the situation significantly. Hans Kurath, Middle English Dictionary, volume 11 (1969) offers this entry for jorī-seil:
jorī-seil n. {Cp. OF jöerie, jüerie joke, jesting, play, etc. also cp. MnE jury-mast.} ?A temporary sail. (1468–70) in Sandahl ME Sea Terms 2 47: j meson seil, j jory saile.
If French words referring to "joke, jest, or play" are indeed the source of jory (which I have not seen argued elsewhere than in Kurath, who is careful to bring them up merely in the form of a comparison), then the underlying idea would be that the a jory-seil is a joke-sail or play-sail, not a real one—and likewise a jury-mast (for example) is joke-mast or play-mast.
Conclusion
I have not found any firm consensus as to the origin of jury (in the sense in which it functions in terms such as jury mast and jury-rigged) beyond the useful observation that it appears to have reached modern English from the Middle English form jory. Nevertheless, there appears to be fairly strong agreement that its origin is not to be found in the hypothetical truncation, at an early date, of the word injury.