Early stages of 'the next big thing'
The earliest instance of the character string "the next big thing" that a Google Books search turns up frames the character string as "the next 'big thing'"—that is, it treats "big thing" as the set phrase and uses next as a simple modifier. From a news item in the Richmond [Virginia] Dispatch (March 9, 1869):
The last war between the Atlantic trunk lines for the trade for and from common points in the West lasted only ten days, terminating on the 27th day of February. The New York Tribune says Vanderbilt won the belt, and makes the Commodore's performance in the premises the next "big thing" to his eighty per cent. dividend on New York Central capital shares.
Here, the expression seems to mean nothing more than "the second-most important holding." The meaning of the phrase at this stage bears little relation to the modern idiomatic sense described in user159691's question.
Somewhat closer to the modern meaning is an advertisement for W.W. Pierce & Co. in the Erie [Pennsylvania] Observer (1869) for several farm implements—a plow, a grain drill, and a seed sower. According to the advertisement, after observing that the Mohawk Valley Clipper Plow "hardly needs a recommendation":
The next big thing is the EMPIRE GRAIN DRILL!
which initially sounds like a thoroughly modern use of "the next big thing," until the ad turns to its next implement:
But the biggest thing yet is the newly introduced and novel Machine entitled CAHOON'S BROADCAST SEED SOWER! This is generating great interest all over the country, and is destined to rapidly become the most popular article of its class.
Evidently, although "big thing" has a very modern sense of "big deal" or "popular favorite" in this advertisement, the wording "the next 'big thing'" as used here just means "the next popular favorite in our lineup."
By 1870, "the next big thing" has taken another step forward and begins appearing in the sense of "the next major planned event." From "Seegers' Ice," in the Newberry [South Carolina] Herald (September 7, 1870):
The chunks of ice for the Herald, came through Messrs. Smiyh & Christian, of this place, who keep champagne on ice, and sudorifics, generally, and who, on this occasion, in their usual gallant style, sent us a bottle of Heidsic, which, with the assistance of the devil, we duly drank to the success of the granite walk, being the next big thing now on the tapis of Newberry.
As noted in a recent answer to an old question about the origin of "on tap," the idiom "on the tapis" means "on the table," "under consideration," or (by extension) "in the offing."
And from an untitled item in the Terre Haute [Indiana] Evening Gazette (December 28, 1875):
The4 New Year's ball, by the Prairie City Lodge, No. 3, at Dowling Hall, will be a big affair.
The ball of the new lodge O. A. U. M., on Friday Evening, is the next big thing on the boards.
These instances, where "the next big thing" is roughly equivalent to "the next big event," appear in multiple places in the late 1870s: in Stroudsburg, Pennsylvania (December 7, 1876) in connection with a political intrigue; in Cairo, Illinois (May 5, 1877), a "baby exposition"; in Newberry, South Carolina again (March 20, 1878), Easter; in Brenham, Texas (July 19, 1878), a "Pomological exhibition"; Hillsborough, Ohio (June 19, 1879), a school commencement exercise; and Memphis, Tennessee (September 2, 1880), a railroad and steamboat excursion. They take us a considerable distance toward the modern meaning, "the newest fad or breakthrough or popular success," and they continue unabated over the next several decades.
Yet another instance that suggests movement in that direction appears in a letter by W.P. Howle, M.D., to the editor of Journal of the American Medical Association (April 28, 1894):
In a friendly chat with one of my medical friends, recently, I asked him why he did not take the Journal of the American Medical Association. His reply was a follows: ""It is too full of big things." There is some truth in this assertion. The Journal is often full of big things; my friend claims that the so-called expert and the specialist are allowed too much space in which to air themselves; that it requires too much effort on the part of the country doctor to digest the "big things." He says the country doctor's time is better spent in reading a few practical truths than in trying to analyze a lengthy article on "Electro-anesthesia and Frequency of Induction Vibration;" "The Esoteric Beauty of One Hundred and Forty-one Histories and Laparotomies for that Disease under Personal Observation." To the specialist these big things are really big, but to the average country practitioner they are too utterly too too. ...
The next "big thing" to which I call attention, is the nostrum vendor. This dare-devil worries me; every mail brings some of his diabolical literature or some of his infernal stuff, "all free gratis and for nothing." I will treat any doctor in the United States to a plug hat who will show me how to stop this fellow from writing me.
Here the writer seems to be using "big thing" in the somewhat sarcastic way that a person might use "big hoopla" or "big to-do," with the implication that "the next big thing" is to some extent a product of popular gullibility and the public's insatiable desire for novelty.
The earliest thoroughly modern 'next big thing'
The earliest occurrence I've found of the expression "the next big thing" in what looks to me to be its thoroughly modern sense of "the new rage in a particular field" is from 1910, in the title of a cartoon by Rube Goldberg from The San Francisco [California] Call (November 22, 1910):
Aeroboxing the Next Big Thing.
The cartoon depicts champion boxer Jack Johnson sailing through the sky in boxing gear strapped to an airplane-like conveyance. Evidently the cartoon was inspired by a comment from the prizefighter: "Jack Johnson predicts that the aeroplane will be used in the fighting game." A second very early instance of "the next big thing" in the relevant sense is from another Rube Goldberg cartoon—this one from the San Francisco [California] Call (September 20, 1912):
The Automatic Life Is the Next Big Thing.
The cartoon anticipates automated shoe stores, hat stores, restaurants, barber shops, and hospitals.
Google Books and Elephind searches turned up several other instances from 1912 and very early 1913. From "What Credit Means to the Farmer," an article on cooperative rural credit associations, in the Abilene [Texas] Reporter (January 10, 1912):
This is the next big thing the commercial organizations of the United States are going to tackle—and accomplish. Why shouldn't the organization in your town be among the first to try it out? What's the use of letting some other town get ahead of you?
From Mysterious Mr. Sabin, serialized in the Molong [New South Wakes] Express (October 19, 1912):
" That is quite true, Mr, Sabin said. "I admit that there are difficulties, but it seems to me that you have overlooked the crux of the whole matter. I have offered you enough to live on for the rest of your days, without ever returning to Europe. You know very well that you can step off this ship arm in arm with me when we reach Boston, even though your man-of-war be alongside the dock. They could not touch you—you could leave your—pardon me—not too honourable occupation once and for ever. America is nob the country in which one would choose to live, but it has its resources —it can give you big game and charming women. It is not Europe, but it is the next big thing. Come, you had better accept my terms!"
From The International Socialist Review (December 1912), in an article by Samuel Ball called "The Next Big Thing":
The phrase, "Life is just one thing after another," rings true. It is even more noticeably true in the activity of the Socialist movement, where one big thing follows another in quick succession, each one bigger and closer on the heels of the preceding one. We have not time to rest up from one season of activity till we are fairly plunged into the midst of another.
First it is the city, then the state campaign. We have not time to catch even a "forty winks" until the presidential campaign is upon us, and now before the smoke of battle has fairly cleared away, we look around for the next big thing to find that already before we knew it, it has established itself and grown to towering dimensions.
The next big thing is the Lyceum. It is already ushered in. As nearly as may be judged, the Lyceum Department has done every thing possible in the way of preparation.
From an untitled item in the Rock Island [Illinois] Argus (January 22, 1913):
The next big thing in England will be the dethroning of the house of lords.
Other instances in which "the next big thing" describes a wave of popular enthusiasm for a real or imagined innovation or money-making scheme, rather than excitement about a discrete scheduled event involve buying property along anticipated main transportation lines near Los Angeles (March 15, 1913), tree crops (November 29, 1913), chemotherapy (December 1914), an expanding copper company (February 16, 1917), funding research to improve U.S. aircraft (July 20, 1919), the Catherine mine (August 9, 1919), automobiles made of cotton (February 3, 1922), wireless radio receivers (April 24, 1922), and anonymous authorship (1925).
'The next big thing' in 1970s popular music and beyond
My own memory of the rise of "the next big thing" was in connection with the music industry, where in the early 1970s record companies and the music press were on alert for "the next Dylan," "the next Beatles," etc., and ultimately settled on the more generically appealing "the Next Big Thing"—sometimes rendered in initial caps—for a band, a singer, or an emerging musical genre. That is the seems to be the gist of Sam Sutherland, "Next Big Thing Fever," in High Fidelity magazine, volume 29 (1979) [combined snippets]:
American popular music is as much a product of fashion as of art. Now that the recording industry is again embracing rock & roll after its year-long affair with disco, that underlying tension between commerce and artistic expression merits particularly close scrutiny. Against a backdrop of this year's severe sales slump, New Wave-influenced rock has made the transition from grass-roots ugly duckling to commercial swan. Executives who found the rough-and-tumble style laughable at its inception have embraced a new faith, fervently hailing the movement as their salvation. While we agree on aesthetic terms, and commend the trade's more creative producers and a&r folk for their sincere support, the motives fueling many of the conversions are somewhat questionable.
Now the music industry, spurred by radio's new willingness to expose this revitalized rock, is applying the same overkill tactics it did to disco. The Next Big Thing is being trumpeted, the bait sweetened by the very cost savings outlined in POP-POURRI last month.
And from a review of David Bowie's Let's Dance, in Stereo Review (1983) [combined snippets]:
"Let's Dance" is Bowie's first album since I980's "Scary Monsters," and, as with all of his records, there's at least a chance that somewhere in these grooves is the Next Big Thing. The surprise is that what's new here is emotion — not the tortured identity crisis of some exile from Mars or the world-weariness of a gigolo but a simple, almost old- fashioned hunger for real love in the real world. The songs on "Let's Dance" work on two levels. First, you can dance to almost everything here.
But whether the music industry's use of "the next big thing" invigorated a usage that had been in continuous since the early 1910s or merely coincided with an older, forgotten tradition, "the next big thing" was a popular expression in the 1910s and 1920s, with essentially the same meaning that it has now. And if any one person deserves credit for popularizing the expression in the 1910s, that person is the great U.S. cartoonist Rube Goldberg, who accounts for two of the earliest definite modern-style instances of "the next big thing" that I found in my Elephind newspaper database and Google Books database searches—dating to 1910 and 1912.