It is reported that Martha Washington said hosting Thomas Jefferson and Mount Vernon was the second-most horrible day of her life, after burying her husband. Do we know what he did during that visit to garner that kind of response.

Next to nothing is known about the meeting first-hand. Jefferson wrote to his daughter Mary Jefferson Eppes about the meeting immediately after it happened, and only had this to say:

"I went yesterday to Mount Vernon, where Mrs. Washington & Mrs. Lewis enquired very kindly after you. Mrs. Lewis looks thin, & thinks herself not healthy; but it seems to be more in opinion than any thing else. She has a child of very uncertain health."

The Mrs. Lewis referred to was Eleanor Parke Custis Lewis, Martha Washington's granddaughter and George's step-granddaughter.

Apparently, the meeting was very brief. The footnote to the letter linked above (taken from The Papers of Thomas Jefferson, vol. 32, 1 June 1800 – 16 February 1801) indicate Jefferson left for Mount Vernon on January 2, 1801, reached the estate the following day, and began his return back to Washington, D.C., before the end of that day. He spent the night at Gadsby’s tavern in Alexandria, Virginia, which is about halfway back. (Mount Vernon is less than 20 miles from Washington, D.C.)

If Martha Washington wrote anything about the meeting, it doesn't survive, because the last of her surviving papers date to 1799, more than a year before the meeting took place.

Washington, D.C.'s National Intelligencer newspaper (something of the Washington Post of its time) did not write about the meeting. Aside from Jefferson's brief comment, everything that is know about it comes from second-hand accounts years later, most thoroughly—but still scantily—reprinted in Ron Chernow's biography of George Washington, Washington: A Life, which I will reproduce here in full, since it's short and just about all you will get on the subject:

"In early January 1801 Jefferson made a pilgrimage to Mount Vernon to see Martha, a visit with an unspoken political agenda. A few weeks earlier it had become clear in the presidential race that Aaron Burr would tie him in the Electoral College, throwing the race into a House of Representatives dominated by Federalists. Jefferson may have thought a well- publicized trip to Mount Vernon would curry favor with Federalist congressmen. If he did, he got precious little thanks from Martha, who fully shared her husband’s cynicism about Jefferson. A friend recalled, “She assured a party of gentlemen, of which I was one . . . that next to the loss of her husband, [the visit] was the most painful occurrence of her life. He must have known, she observed, that we then had the evidence of [Jefferson’s] perfidy in the house.”

"Taking the high road in his first inaugural address, President Jefferson named Washington as “our first and greatest revolutionary character, whose preeminent services had entitled him to the first place in his country’s love.” Martha Washington was not assuaged. “Her remarks were frequently pointed and sometimes very sarcastic on the new order of things and the present administration,” wrote Manasseh Cutler. “She spoke of the election of Mr. Jefferson, whom she considered as one of the most detestable of mankind, as the greatest misfortune our country has ever experienced. Her unfriendly feelings toward him were naturally to be expected from the abuse he offered to Gen. Washington while living, and to his memory since his decease.”"

The "friend" quoted in the first paragraph was John Cotton Smith, from Correspondence and Miscellanies of the Hon. John Cotton Smith. Smith was a Federalist politician from Connecticut who was serving in the U.S. House at the time; he would later serve as Lt. Governor and then Governor of his home state. The recollection was published in 1847, though undated. Smith died in 1845, so it wasn't written any later than that, though maybe not much before, as some of the other writings in the same chapter ("Miscellany") were dated to 1844. Nonetheless, it seems to be a collection of undated memoirs Smith wrote on various topics over a period of years, and the full relevant quote is actually more revealing than what Chernow reprinted. After chiding Jefferson for various political reasons, Smith wrote:

"If, in this brief survey, we look in vain for...[a] high sense of honour which should characterize a successor of Washington, what shall we say of [Jefferson] as a subject of the moral government of Jehovah? Think of his visiting the tomb of Washington, and shedding crocodile tears over his remains; a transaction evidently intended, for it was extensively published, to favour his election to the presidency, but which was truly distressing to Mrs. Washington. She assured a party of gentlemen, of which I was one, at her house in January, 1802, that, next to the loss of her husband, it was the most painful occurrence of her life. He must have known, she observed, that we then had the evidence of his perfidy in the house."

The second quote that Chernow pulled from was from Manasseh Cutler, a clergyman and another U.S. House member aligned with the Federalists and friendly to the Washingtons. The quote comes from a diary entry of his, dated a year later, on January 2, 1802 - perhaps both he and Smith had been in attendance at the same time and heard Martha recount the episode a year after the fact. The book The Washingtons: George and Martha, "Join'd by Friendship, Crown'd by Love" by Flora Fraser adds an additional line, as seen in the link to Cutler's diary above, so that the relevant quote reads in full:

"We were all federalists, which evidently gave her particular pleasure. Her remarks were frequently pointed and sometimes very sarcastic on the new order of things and the present administration. She spoke of the election of Mr. Jefferson, whom she considered as one of the most detestable of mankind, as the greatest misfortune our country has ever experienced. Her unfriendly feelings toward him were naturally to be expected from the abuse he offered to Gen. Washington while living, and to his memory since his decease."

The context of the meeting is important. While the National Intelligencer did not write about the meeting itself, they were writing frequently about Jefferson in December, January, and February 1800-01, because he had just been elected president—sort of. This was the election where Jefferson and Aaron Burr had tied in the Electoral College, and under the original provisions of the U.S. Constitution, there were not separate votes for POTUS and VPOTUS. And in the case of a tie, the election got thrown to Congress to break the tie. And Congress was made up of a Federalist majority. They were an outgoing, lame-duck majority, but they were going to pick the next President.

This is why John Cotton Smith, and Ron Chernow, wrote what they did. Jefferson made his visit to make good with the Federalists, in hopes of getting some good publicity and turn the vote toward him. In fact, in the very next line of the letter Jefferson himself had written on the meeting, quoted above, he writes to his daughter what he's heard the vote total is expected to be when the election reached Congress (73 for him, 64 or 65 for Burr).

Yet, it still wasn't a sure thing. Jefferson was the leader of the Republican/Democrat/Anti-Federalist party, and the Federalists despised him. The only thing working in his favor was that many Federalists distrusted Burr even more. Jefferson's treatment of Washington both in life and in death was a particular sore point. Jefferson hadn't even attended Washington's funeral, though Jefferson defended that by deeming it inappropriate considering the bad political blood between them. But now, here he was, at Martha Washington's doorstep, more than a year after Washington had died, shedding "crocodile tears over his remains" in order "to favour his election to the presidency" as John Cotton Smith put it.

I won't go over what led up to this, since you already mentioned Jefferson and Washington being "frenemies". Rest assured, none of this was lost on Martha Washington. She was well aware of the role Jefferson had played in getting the press to trash Washington in public during his presidency. As Patricia Brady wrote in Martha Washington: An American Life:

"Besides the admirers [who she had kind words for], Martha also tenaciously remembered those who had hurt her husband. When Thomas Jefferson was elected in 1800, she commented freely and acidly on his presidency. She never forgave the former intimate who had wounded her husband so cruelly for political ends."

As for anything specific Jefferson said or did during the visit, it has not been preserved. There may not have been anything in particular that happened. But the visit in general, at least, and cynical timing of it, evidently bothered Martha Washington, according to the little we know of it second-hand. She did not appreciate being used as Jefferson's political pawn, nor as a possible advocate for his election over Burr.

While someone else may come up with some more info, the above is pretty much all that's known of the meeting. I consulted the following sources, and they either reprinted some or all of the quotes above, or else they didn't mention the meeting at all.

Washington: A Life by Ron Chernow

The Washingtons: George and Martha, "Join'd by Friendship, Crown'd by Love" by Flora Fraser

Martha Washington: An American Life by Patricia Brady

Jefferson and the Ordeal of Liberty and Jefferson the President: First Term, 1801-1805 by Malone Dumas

Washington (abridged) by Douglas Southall Freeman

Washington: The Indispensable Man by James Thomas Flexner

American Sphinx: The Character of Thomas Jefferson by Joseph J. Ellis

His Excellency: George Washington by Joseph J. Ellis

Thomas Jefferson and the New Nation; a Biography by Merrill D. Peterson

EDIT: Thanks for the platinum! And gold!