Early instances of 'remake' as a noun in Australia
The earliest use of remake as a noun that Google Books finds is from Australia, where the term appears in documents related to making alterations to military uniforms. From testimony of Sergeant J. Lyttleton, a master tailor at Victoria Barracks in Sydney in "Court of Inquiry—Supply and Issue of Clothing, Permanent Artillery" (September 17, 1888), in Proceedings of a Court of Inquiry and General Courts Martial... in the Case of No. 95, Brigade Quartermaster-Sergeant (Now Gunner) William Webster, N.S.W. Artillery (1889):
Q. How was it determined that 18s. for a remake should always be charged when found the 1s. would not cover?
A. I took over the custom of 1s. or 18s. which was prevailing when I joined.
Q. Who determines the amount of alteration required by a garment?
A. The master tailor.
Q. Is there any check on your determination. Suppose you settled a remake which was not perhaps quite necessary, who could stop you?
A. My decision has to be acquiesced in by the Brigade Quartermaster-sergeant, Mr. Webster.
This same witness had mentioned "a remake" in the sense of garment alterations in testimony as early as July 4, 1888.
But a considerably earlier instance (also from Australia) evidently involves a "reef" (stratum of rock) that reappears in a different location from the one where it was originally discovered. From "The Great Britain Company" in the Bendigo [Victoria] Advertiser (March 8, 1865):
The stone which has been got from the company's ground within the last few days, and which we were shown yesterday, looks very well. The claim which the company holds is situated near Kangaroo Flat, and comprises 14½ acres, held on three separate leases. The ground includes a portion of the Caledonian line of reef ; also the Wallace line. For some time past the company has bad three tribute parties at work on the Caledonian line, and one tribute party on the Wallace line. The reef from which the specimens were obtained is five feet and a half in thickness, and appears to be what is termed a "remake" of a number of spurs which were worked on the two lines. Should the whole reef with its great thickness turn out "equal to sample," there is a good prospect for the shareholders.
This usage seems to have persisted for several decades at least, since the term remake appears again in multiple Australian newspaper accounts of gold and quartz prospecting. From "Mining Intelligence," the Bendigo Advertiser (June 27, 1871):
The success of the Paris Reef Company at Break-o'-Day has caused considerable attention to be diverted to that part of the country, and already ground north and south has been taken up and companies formed. The old Break-o'-Day claim, which once yielded such good results, and gave its shareholders thousands in dividends, has been taken on tribute from the Bendigo and Melbourne Company, who own it, by a party of Melbourne men, and it will be worked in conjunction with other leases taken up adjacent. There is no doubt whatever but that at greater depths a remake of the golden reef will be met with: that is if we judge from the experience of other companies in this district. But, no doubt, this will require some money to be spent in prospecting.
From "The Gold-Fields," in the Sydney Morning Herald (January 20, 1872):
The last body of stone had cut out on the dip in the usual way in this field, but, contrary to the usual formation of a break, showed no indications of a remake of the reef in the foot wall, as is usually looked for in these cases. It therefore became rather a puzzle to ascertain what had become of the reef, seeing that, as all skilled quartz miners well know, there is no such thing known as a reef dying out or coming to an end.
From "Mining Notes," in the [Melbourne, Victoria] Argus (April 1, 1882):
Grant's Amalgamated, March 25. ... No. 3 Shaft.—South drive extended 3ft 6in ; ground hard , reef bearing towards the east too much, and poor. I caused the west side to be blasted into to-day, and found a remake of the reef 1ft ; tried a prospect, which was payable.
From "Mining Intelligence" in the Carcoar [New South Wales] Chronicle (June 24, 1882):
Grant's Amalgamated G.M. Co.—Mr. T. G. Williams reports: ... No. 3 shaft.—We have commenced cleaning the mullock up, but in consequence of the rain, I have put the men to cross-cut to the west, and on blasting we found a remake of stone about 9in in width containing good payable quartz.
From an untitled item in the Bendigo Adveriser (January 18, 1890):
The Nil Company (Raywood) have obtained 72 ounces of gold, including the copper plates. At the intermediate level the manager has come upon some stone this week, from which several specks of gold were obtained, and it is thought this may be a remake of the reef, which was successfully worked from No. 4 level. The stone is of a very solid character, encased in slate, and showing plenty of mundic and galena.
Additional instances of remake in the stratigraphical sense appear in the [Adelaide] South Australian Register (December 22, 1891), the [Perth] West Australian (September 4, 1897), and the Kalgoorlie [West Australia] Miner July 12, 1912).
Early instances of 'remake' in a theatrical sense
The earliest instance that Elephind identifies of remake in the sense of "update an entertainment" is from "'What Every Woman Knows' Praised by New York Critics," in the [Maysville, Kentucky] Public Ledger (March 8, 1922), reprinted from Variety:
It is likely that a younger generation will see "What Every Woman Knows" as a picture although there will be a sprinkling — plentiful at that — of those who remember the legitimate version of the play. The distinct feature of the picture is that, although it is based on a great play, it stands up as a picture regardless of its stage fame. ..... "What Every Woman Knows" has not twisted continuity or a remake into something "modern." [J. M.] Barrie in this pictures is undiluted Barrie. His text is used freely and with discretion, and in not a single instance is it misapplied. It retains, therefore, a great deal of its original flavor, and that is very satisfactory.
And from "Behind the Screen in Hollywood," in the Madera [California] Mercury (July 15, 1925):
[Oscar] Price controls all the famous stories made once by “Triangle" [Film Corporation] some years ago, with stars like Fairbanks, Bill Hart, Charlie Ray, Lillian Gish, etc. Price has completed a remake of Doug's greatest success, "Manhattan Madness" with Mr. and Mrs. Jack Dempsey starred, and is now using Glenn Hunter of "Merton of the Movies" fame, in a series of the stories that made Charlie Ray famous.
From James Hamada, "Brenon Directed Film at Hawaii; 'Madness' Comes to the Prince," in the [Honolulu, Hawaii] Nippu Jiji (December 28, 1932):
Coming back to "The Girl of the Rio" at the Hawaii, this is a re-make of the silent picture, "The Dove," that Norma Talmadge made in 1928. It isn't quite as good as the silent—in fact, very few talkie re-makes of silent successes have been successful so far. Its principal interest lies in Dolores Del Rio and Leo Carillo.
From "Film Artists Claim Freedom," in the [Brisbane, Queensland] Sunday Mail (January 15, 1933):
The classic example of a screen star demanding a say in the choice of & story, of course, is that of Marlene Dietrich, who, with Director Josef von Sternberg, "walked out" after a quarrel with Paramount. There have been other cases, Janet Gaynor winning her struggle to get away from the romantic sob-stuff part, and Ann Harding was allowed a remake Westward Passage in accordance with her ideas.
Additional instances of remake appear in issues of the Nippu Jiji from February 27, 1933, May 16, 1933, and December 23, 1933, in all three instances involving talkie versions of earlier silent films. A dozen or so additional matches for "remake" in the relevant sense appear in newspapers in 1934 and 1935.
The evidence from Elephind searches suggests that remake as a noun was used in mining terminology in Australia for at least five decades, beginning no later than 1865; and that it was used in the sense of garment alterations, again in Australia, for an indeterminate amount of time, starting no later than 1888.
In the show business sense of releasing a new version of an older movie or theatrical performance, the word seems to date back to at least 1922 in the United States.