The earliest usage I can track down is a quote in a 1986 Wall Street Journal article, by an IBM executive (just a few months earlier than the quote @Xanne found in the NYT, also about IBM)1:
[Events for non-marketing reps] aren't as lavish, and fewer people are invited. "That's mainly because (others) don't have their skin in the game the way the marketing rep does," says Doris Isaacs, an IBM director of systems engineering. (Dennis Kneale, "Working at IBM: Intense Loyalty in a Rigid Culture," Wall Street Journal (1923 - Current file), Apr 07, 1986, pp. 27, ProQuest Historical Newspapers: The Wall Street Journal.)
This would seem to have been a not-unknown phrase in the computer world; in 1997 we have:
In the old days, a scant decade ago, systems integrators would characterize their stake in a large deal by proudly claiming they had "skin in the game." By this they meant they had made a sizable investment: buying the customer's assets, transferring the customer's employees to its staff, maybe transferring software licenses onto their books and so on. (Susan Scrupski, "Turning Contracts into Partnerships," Computerworld, July 28, 1997. LexisNexis Academic.)
The phrase also probably got a boost in the mid-nineties from maverick billionaire presidential candidate Ross Perot, who used the phrase several times during his campaign. For example:
In the C-SPAN interview, Perot said he would spend $50 million to $100 million of his own money on a campaign, although he said he would encourage his supporters to contribute $5 "because I want them to have skin in the game." (Mark Stencel, "Perot Weighs Candidacy; Presidential Race Entry is `Up to the People'," The Washington Post (pre-1997 Fulltext), Mar 23, 1992, pp. a01, US Newsstream.)
Another contemporary source quotes Perot:
[I]f he runs, Perot says, he would invite "ordinary folks" to put $5 each into it, so they would have a stake. . . . "I proposed that we have a war tax before we go into this so that all of us had 'skin' in the game," he says. "Isn't it bizarre that the only heroes from this [first Iraq] war are generals and politicians?" (John Dillin, "Possible Presidential Bid by Perot Is Seen Posing a Threat to Bush," Christian Science Monitor, March 24, 1992.)
And during his 1996 campaign:
Rather than spend more money of his own, Perot has decided to take the taxpayer money, explaining, as he put it on "Good Morning America" yesterday, that he wanted to "make sure the American people get involved" in his campaign effort. Speaking on CNN's "Larry King Live" Sunday night, Perot said, "In talking to all our members, they wanted to put some skin in the game. We're asking them to make small contributions." (Ruth Marcus & Donald P. Baker, "CHANGE WORTH $29.2 MILLION FOR PEROT," Washington Post, August 20, 1996.)
Perhaps significantly, Perot got his start as one of those high stress, high pay IBM salespeople, and went on to make his fortune in the computing industry (Wikipedia).
From there, usage takes off quickly, as attested in the Ngram cited in the question.
But where did the phrase actually come from? The Perot quotes are strongly suggestive of a connection between money, something like an ante, and "skin". I suspect that the phrase may have originated with gambling, and the slang use of the word skin to mean money or (more specifically) US dollars. From the OED:
- fig. U.S. slang. A dollar; = frogskin n. (a) at frog *n.*1 and adj. Compounds 2a. ("skin, n." OED Online. Oxford University Press, March 2017. Definition 6).
This usage for skin is attested from 1834 through 1997. That last quote is in reference to golf, where the term is still used to refer to a betting version of the game. From Golf for Dummies:
Skins: Players bet a certain amount each hole – a skin – but if two tie, all tie, and the money is added to the pot for the next hole. If a foursome plays skins and no golfer beats the other three on any hole, you can wind up with five or six or even more skins riding on a later hole. To win at skins, relax early in the round and focus on playing your best when the chips (well, skins) are down. And don't be afraid to take risks. Remember, you have to win the hole outright to claim the skins. (Gary McCord, Golf for Dummies, 2012.)
Thus if a player hasn't anted up—bet a "skin"—in the golf game, he or she may not be as concerned about playing their very best.
One more circumstantial piece of evidence: Golf was at one time part of the insular IBM culture described in that first article.
By the late 1960s, IBM had become the apex of how companies treated workers and thought of their roles in society. Its culture was called "cradle to grave," meaning if you got in, they’d take care of you. There were lavish carnivals for workers and their families. Around the country, there were country clubs and golf courses where workers at all levels could play for virtually nothing. Casperkill Golf Club in Poughkeepsie, New York, is the site of a former IBM country club. IBM sold it more than a decade ago, and you can still find retired IBMers grumbling about the changes. (Dan Bobkoff, "IBM: when corporations took care of their employees," Marketplace, June 13, 2016.)
So, not a definitive answer, but a conjecture: once upon a time, somewhere between the 1960s and 1980s, IBM employees regularly played a form of gambling golf. Having money on a hole, called "skin" in the game, was known to make that hole's play matter more to the player. Thus having skin in the game became an IBM corporate catchphrase for being personally invested in an endeavor. The phrase gradually leaked out into associated industries, and eventually caught on with the general public (and maybe especially those finance execs and politicians who might be expected to still play "skins" golf today).
1Unfortunately, nearly all of these sources are behind pay-walls; hopefully, the citations are full enough that folks can find them (perhaps through their local library) if desired.