The limitations of Ngram data
Ngram can't give you an accurate plotline for "anybody's guess" because it construes anybody's as anybody - s. Despite this failing, Ngram does return matches for "anybody's guess"—I'm just not sure how reliable those results are.
The two earliest Google Books matches for "anybody's guess" are from the early 1920s. From "Annual Report of the Director of the Census to the Secretary of Commerce for the Fiscal Year Ended June 30, 1921" (1921):
The increase of nearly 150 per cent in the total value of products between 1914 and 1919, as shown by the manufactures census of 1920, does not, of course, represent any corresponding increase in the volume of production. So far as these value figures are concerned, the actual increase of production is anybody's guess, the question being one which the census has not answered.
And from class of '10 alumni notes in Princeton Alumni Weekly, volume 24 (March 26, 1924):
Mort Easton gives his business address as in care of the National Mayonnaise Machine Co., 207 Pacific St., Brooklyn, N.Y. The Secretary lives in Brooklyn, so the address is intelligible to him, but as to the nature of the business, anybody's guess goes. In these days of radio and spiritualism, why shouldn't mayonnaise be made by machinery?
But the next match is from the Gas Age-Record (January–June 1929):
The Future—"Anybody's Guess"
So it is not surprising to hear people say that the future is "anybody's guess." Th nation's greatest leaders of finance are not at all in accord respecting the handling of basic problems.
That's rather a long gap for a phrase that evidently was popular enough in 1929 to be given the "people are saying" treatment. An even bigger red flag involves matches for the apostrepheless phrase "anybodys guess." Such matches presumably involve accidental omission of the apostrophe in anybody's, so you might expect them to be rare and later than instances of "anybody's guess." But a Google Books search finds three matches for "anybodys guess" between 1916 and 1922. From "Talked Among Brokers" in United States Investor (April 15, 1916):
I hear numerous opinions accounting for the present stagnated condition, but as a matter of fact, a man who has been in Wall Street for the last 30 years, and who has been connected with many important deals in that time, said to me this week, "It seems almost as if it were anybodys guess. The situation as presented to-day somewhat parallels that of about a year ago, except that where prices were abnormally low then, many of them are very high to-day.
From "Greeley [Colorado]" in Bulletin [of] Agricultural, Industrial and Mining Conditions (April 30, 1920):
There is some apprehension as to whether we will get our reservoirs filled, which dpends [sic] largely on the spring rains to help bring down the water from the melting snow. It is anybodys guess as to what the results this season will be.
And from The Cooperative Manager and Farmer: Devoted to the Interests of the Grain, Flour, Feed and Coal Trade (1922) [combined snippets]:
Wisconsin, Minnesota, eastern Iowa and northern Illinois have a big crop of well dairy-bred veals to cash and during the spring period of heavy supply the calf market is anybodys guess. There is this to be said about cattle feeding, hereafter steers must be bought right and fed economically to make the finality of the process satisfactory. It is not a game for an amateur, or the careless. The period of wide profit margins has passed.
U.S. Library of Congress data
These results seem highly implausible—and a check of the U.S. Library of Congress's Chronicling America database of historical newspapers confirms their inaccuracy. Chronicling America finds dozens of matches from 1922 and earlier for "anybody's guess." One very early match is from "Correspondence of the Republicans," in the [Plymouth, Indiana] Marshall County Republican (August 2, 1860):
Republicans are confident, and promise the most glorious of victories, but they haven't got a solitary figure to show for it. Now this won't do. We have been mixed up in politics too long to trust anybody's guess before a plain, correct, full list of voters. We want no guessing when we can have the fact, as fixed and absolute as arithmetic can make it.
But a match from 1897 seems much closer to the modern sense of "anybody's guess." From "Wheat Prospects" in the Indianapolis [Indiana] Journal (February 24, 1897):
The price of wheat receded 1 to 1½ cents per bushel yesterday, and it is now about 12 cents per bushel below the highest point touched during the present crop year. It is anybody's guess whether it will recover lost ground, remain around the present level or go lower before the crop year shall end.
And from "Bought by the Northern Pacific," in the Little Falls [Minnesota] Weekly Transcript (January 19, 1900):
At present the Brainard & Northern is operated independently, but with close traffic arrangements with the Northern Pacific, and it might be anybody's guess that the road will eventually become a part of that system.
Chronicling America finds 23 additional confirmed matches for "anybody's guess" in the relevant sense from the decade 1901–1910. So it appears that the usage of "anybody's guess" to mean "not known and perhaps not knowable" dates from the very end of the nineteenth century and had become fairly popular in the United States by 1910.
The only hint that Chronicling America gives regarding the source of this expression is the fairly common use—during the late 1800s and early 1900s—of the phrase "Can anybody guess?" as a teaser question in news reports, count-the-beans-in-a-jar contests, and the like. It may be that "anybody's guess" emerged in the first instance simply as an extension of this common question.