Generally speaking, the legend originated in the mid-19th century in the United States. Who invented it or exactly when they did is a bit harder to pin down. It was likely being told orally for some time before it was recorded in surviving literature.
According to Christmas In America: A History by Penne L. Restad, the first known direct reference to Santa Claus living at the North Pole comes from an illustration by Thomas Nast, published in the December 29, 1866, edition of Harper's Weekly magazine. You can see the original illustration here, and a lower resolution version but all on one page here. If you look closely, northeast of "Claus" in large print, you'll see the words "Santa Claussville, N.P." inside the circular border. The "N.P." is thought to stand for "North Pole".
The earliest instance yet discovered of "North Pole" spelled out in full in connection with Santa Claus is in the poem "Santa Claus and His Works" by George P. Webster, which was published in 1869, incidentally with drawings by Nast. The poem contains the lines:
"His home through the summer months, you must know,
Is near the North Pole, in the ice and snow."
"I told you his home was up North by the pole
In a palace of ice lives this happy old soul."
By the 1870s, this part of the mythology was common knowledge. As some examples, the December 25, 1873, edition of the Richmond, Virginia, Daily Dispatch newspaper listed an undeliverable letter held at the local post office addressed to "Santa Claus, Greenland, North Pole".
The Boston-based children's magazine The Nursery published a letter in 1874 from a boy in Colorado that read in part: "If we did not live so very far from the North Pole, I should ask Santa Claus to bring me a donkey."
The December 26, 1874, edition of the Brooklyn Daily Eagle newspaper included a lengthy article entitled "Christmas - How the Day Was Observed In Brooklyn", which contained the following:
"Thursday evening was an occasion of unusual interest to the children, teachers, superintendent, pastor, and congregation. The children had received a letter dated July 4, from the North Pole, that Santa Clause would really visit them on the night before Christmas, and so they all turned out in full force, several hundred of them."
Thomas Nast himself made the "North Pole" location of Santa Claus more explicit in later drawings published in the 1890 collection Thomas Nast's Christmas Drawings for the Human Race. Some of the drawings were reprints of his earlier work for Harper's Weekly, and some were new, though I think all the "North Pole" references were from the 1880s or new to 1890, and certainly not older than 1866. In a drawing entitled "Santa Claus's Route", two children study a map with the words "North Pole" on it. In a drawing entitled "Christmas Post", a child drops a letter into a mailbox, addressed to "St. Claus, North Pole".
That all said, Nast probably didn't originate the idea. According Restad's book, "Exactly how [Santa's home at the North Pole] came to be is not clear." Restad recounts that John Pintard, the founder of the New-York Historical Society who was instrumental in promoting the Santa Claus mythology in its early stages, wrote a letter from his Manhattan home in December 1829, claiming that Santa Claus "arrived from Spain, in a Dutch ship filled with toys, and docked each Christmas Eve in New York". The Dutch Sinterklaas myth still has it that he arrives annually from Spain, and Restad may be making an assumption here of what Pintard believed. Reading the original, Pintard writes:
"The annual ship from Amsterdam, Alder lievest Vrow (best beloved wife) with St. Class & his presents for good children has arrived off the Hook & a Steamboat is sent down to tow her up, so as to prevent any disappointment to night..."
It would seem Pintard's belief was that Santa made his annual pilgrimage from from Amsterdram to New York City, and Restad is assuming that Pintard believed this followed Santa's trip by boat every year from Spain to Amsterdam. In any case, neither the North Pole nor a northerly homeland is referenced at this date.
But there was a northerly homeland referenced at least as early as 1859. As Restad notes, the 1859 book History of the City of New York by Mary Louise Booth contains the passage:
"To the children, [Santa Claus] was a jolly, rosy-cheeked little old man, with a low-cronwed hat, a pair of Flemish trunk-hose, and a pipe of immense length, who drove his reindeer sleigh loaded with gifts from frozen regions of the North over the roofs of New Amsterdam for the benefit of good children."
Here, Booth is writing, perhaps apocryphally, that these were the beliefs in old Dutch New York in the 1600s, that Santa Claus was a character who appeared from his home in the "frozen regions of the North". There is no evidence that these were the beliefs of the time, and Pintard's version was probably closer to what was believed in that period. But considering that Booth was trying to write a true history of New York as of 1859, it appears that this Northern homeland was understood by her not to be a recently-invented tradition.
Another Northern reference, but not to the North Pole explicitly, is found in an illustration published in the December 26, 1857, edition of Harper's Weekly (this one predates Nast's tenure with the magazine). "The Wonders of Santa-Claus" depicts Santa sitting on an ice throne, and starts the accompanying poem with:
"Beyond the ocean many a mile,
And many a year ago,
There lived a wonderful queer, old man
In a wonderful house of snow."
Similarly, the December 16, 1857, edition of the Geneva Courier in Geneva, New York, reprinted an item from the Albany Journal, which said in part:
The Rumored Failure of Santa Claus,—The North wind came down last night from Lapland blew positively that the rumored suspension of the Christmas Saint was wholly untrue. Our informant says that he followed the universally loved old fellow into his reindeer stable...[T]he whole sky back of the Santa Claus mansion was suspended thick with every conceivable thing to gladden the hearts of children. Mr. Wind, however, seemed to feel that the Sledge was not as large at that in use of late years..."
It would seem, by these accounts, if the North Pole hadn't been settled on specifically by the 1850s, then at least a very far northern locale had been. (Interestingly, a February 26, 1876, article in the Rockland County Journal published in Nyack, New York, echoes the Lapland reference: "After Christmas-time come letters to Santa Claus at the North Pole, or Kriss Kringle in Lapland...")
There are references before 1830 by Washington Irving and (famously) by Clement Clark Moore, and by others, of Santa delivering toys through chimneys, conveyed in a sleigh by one or more flying reindeer. Reindeer being an animal associated with the north, it's quite possible that the myth of a northern homeland of Santa Claus had been orally communicated by New Yorkers and other Americans for quite some time before Booth, Nast, and Webster wrote down the myths in the 1850s and 60s.
Restad proposes, as did an article published by the New York Times, that a series of well-publicized expeditions to the Arctic in the 1840s and 50s captured the public's attention and it was perhaps because of this that the North Pole myth gained popularity during this era. In any case, the surviving literature points to an origin for the North Pole myth no later than the mid-1860s, and was commonly referenced by the early 1870s. There are mentions of a Northern homeland for Santa by the 1850s, if not explicitly the North Pole, though the legend may have been communicated orally by that point.
TL;DR: Santa Claus making his home in a far North location dates at least to the 1850s. The North Pole legend may have been communicated orally during that period, and perhaps even earlier. The earliest surviving mention in print of Santa's home in the North Pole is probably from an 1866 illustration by Thomas Nast, which contained the text accompanying his workshop "Santa Claussville, N.P." More explicitly, the North Pole is mentioned in the 1869 poem Santa Claus and His Works by George P. Webster. By the early 1870s, many explicit mentions of Santa's North Pole home are found in surviving literature across the United States.