It would seem extremely unlikely that Thomas Jefferson would run for a third term as President in 1808. Jefferson, after all, was the real intellectual author of the no-third-term tradition. Washington had retired after two terms, and had thus set a precedent, but Washington had never indicated that it would be improper for another President to run for a third term. Most likely he was just weary of the presidency, and especially of the vicious Republican attacks on him since Jay's Treaty, and anxious to get back to his plantation. Jefferson, OTOH, was worried from the beginning about the Constitution allowing a President to be re-elected indefinitely, and explicitly intended his retirement after two terms to prevent future Presidents from seeking a third term. As he explained in a letter to John Taylor on January 6, 1805, just before the beginning of his second term:
"...My opinion originally was that the President of the U.S. should have been elected for 7. years, & forever ineligible afterwards. I have since become sensible that 7. years is too long to be irremovable, and that there should be a peaceable way of withdrawing a man in midway who is doing wrong. The service for 8. years with a power to remove at the end of the first four, comes nearly to my principle as corrected by experience. And it is in adherence to that that I determined to withdraw at the end of my second term. The danger is that the indulgence & attachments of the people will keep a man in the chair after he becomes a dotard, that reelection through life shall become habitual, & election for life follow that. Genl. Washington set the example of voluntary retirement after 8. years. I shall follow it, and a few more precedents will oppose the obstacle of habit to anyone after a while who shall endeavor to extend his term. Perhaps it may beget a disposition to establish it by an amendment of the constitution. I believe I am doing right, therefore, in pursuing my principle. I had determined to declare my intention, but I have consented to be silent on the opinion of friends, who think it best not to put a continuance out of my power in defiance of all circumstances. *There is, however, but one circumstance which could engage my acquiescence in another election, to wit, such a division about a successor as might bring in a Monarchist. But this circumstance is impossible.* While, therefore, I shall make no formal declarations to the public of my purpose, I have freely let it be understood in private conversation..." (Emphasis added)
http://www.let.rug.nl/usa/presidents/thomas-jefferson/letters-of-thomas-jefferson/jefl167.php
Note, though, that Jefferson does leave a loophole: if division about his successor might bring in a "Monarchist"--by which he presumably means Federalist--he might run again. (Jefferson regarded the Federalist leaders as monarchists at heart. It is true that in his first inaugural address he said "We are all Republicans, we are all Federalists" but by this he simply meant that most rank-and-file Federalist voters were republican at heart, and might be won over to the new administration.) It is understandable why Jefferson in 1805 should regard this as a very unlikely, indeed "impossible" contingency. The Federalist Party had been shattered, was a hopelessly outnumbered minority in Congress, and could not even carry Massachusetts in the 1804 presidential election. The Federalists' last hope of exerting influence on presidential elections--by making the Republican vice-presidential nominee President--had been removed by the Twelfth Amendment.
Jefferson was faithful to the principle he expressed in the Taylor letter. When in 1806-7 the legislatures of Vermont and several other states urged him to be a candidate for a third term, he firmly declined. His December 10, 1807 reply to the Vermont legislature is "his best-known polemic against the third term" (Charles W. Stein, *The Third-Term Tradition,* p. 38):
"That I should lay down my charge at a proper period, is as much a duty as to have borne it faithfully. If some termination to the services of the chief magistrate be not fixed by the Constitution, or supplied by practice, his office, nominally for years will, in fact, become for life; and history shows how easily that degenerates into an inheritance. Believing that a representative government, responsible at short periods of election, is that which produces the greatest sum of happiness to mankind, I feel it a duty to do no act which shall essentially impair that principle; and I should unwillingly be the person who, disregarding the sound precedent set by an illustrious predecessor, should furnish the first example of prolongation beyond the second term of office.
"Truth, also, requires me to add, that I am sensible of that decline which advancing years bring on; and feeling their physical, I ought not to doubt their mental effect. Happy if I am the first to perceive and to obey this admonition of nature, and to solicit a retreat from cares too great for the wearied faculties of age."
http://teachingamericanhistory.org/library/document/letter-to-the-legislature-of-vermont/
And yet...the political situation by 1808 was very different from that of 1805. The Federalists had undergone something of a revival in the Northern (especially New England) states because of the unpopularity of the Embargo. True, they were still too weak to have any prospect of electing one of their own as President, but the "division about a successor" among the Republicans which Jefferson had mentioned in 1805 would at least give the Federalists a chance to influence the election by supporting Monroe or Clinton instead of Madison. If Jefferson was sufficiently worried that the next President might owe his election to "monarchist" (Federalist) support, might he not think that something like the "impossible" exception he had mentioned in his Taylor letter had materialized, and that this might justify him in seeking a third term? If he ran, he would doubtless have won; I agree with Claude Bowers that "had Jefferson himself consented to a third term he could not have done worse [than Madison in 1808], and he might reasonably have done better" (quoted in Stein, p. 42). What would a third Jefferson term have been like? Does the War of 1812 come on schedule, and is the US any better prepared for it than under Madison in OTL? And does the prospect of Federalists and dissident Republicans capitalizing on anti-war feeling even lead Jefferson to run for a fourth term? (Yes, he will feel bitter about apparently vindicating his own prophecy that the right of re-election could lead to something like lifetime presidencies, but will think he had no choice...)