Talk Elections

State them

James Madison, John Adams, George Clinton, Alexander Hamilton, John Jay, Thomas Jefferson, and Rufus King seem most likely.

Some serious contenders for the role ...

Thomas Jefferson - Obvious reasons
John Adams - Obvious reasons
John Hancock - Longest serving President of the 2nd Continental Congress
Cyrus Griffin - Last President of the 2nd Continental Congress
Ben Franklin - Old but the most respected man on the continent
Sam Adams - One of the most involved members of the revolution since the beginning
Roger Sherman - Only man to sign all 4 of the "Great Papers"

Most likely it would be Adams.  He was a respected figure and well known amongst the people who would have been chosen as Electors as a deep thinker on what it takes to run a republic.  That said, while he was an excellent 2nd President, I don't think he would have been a good 1st President.

Adams is by far and away the frontrunner in any scenario where Washington declines the presidency. He was objectively among the mosts influential members of the Second Continental Congress — arguably the most influential, as Chairman of the Board of War and de facto floor leader for the pro-independence faction — had established himself as a leading voice on constitutional theory with Thoughts on Government and his authorship of the Massachusetts Constitution, and enjoyed a prominent and distinguished diplomatic career besides. In actual history, his reputation was considered so great as to potentially threaten the unanimity of Washington's election, and had Washington refused to be considered, there is no question that Adams could have mustered a majority of the electors.

Apart from Adams, the other possible choices are either too old or too young, lack the necessary national reputation for election, or were uninterested in the presidency. Madison and Hamilton fall into the first category — they had only just risen to national attention with the Federalist Papers — as do Franklin and John Hancock. George Clinton is a potential contender, but probably no more than that. Jefferson in 1788 was not yet the leading voice of American liberalism he would become; his national reputation was largely an outgrowth of his time as Washington's Secretary of State, and while we revere him for it today, writing the Declaration of Independence was not a landmark achievement in the popular imagination until the Republicans made it so during the campaigns of 1796 and 1800. Sam Adams seems like a logical choice, but would suffer from the rule prohibiting electors from voting for two candidates from their home state, as the Massachusetts electors would all but assuredly be backing John. Patrick Henry, John Jay, and John Rutledge all probably win votes as well.

If Adams does not win a majority, it's very possible that the election would be thrown to the House, where the states would then have had the choice of the five highest polling candidates. The delegates to the Philadelphia Convention considered this the likeliest outcome of most presidential elections (with Washington serving as a rare example of a candidate with sufficient national support to build a majority), viewing the electoral college as effectively a series of nominating caucuses that would send candidates to the real vote in Congress. Personally, I think this the second-most-likely outcome in lieu of an Adams majority.

Anthony Wayne, Frank Knox or Artemas Ward were all generals who had become quite popular by 1888 are good choices for the kind of "Populist" president. John Jay or Philip Schyuler were prominent statesmen who ran anyways, so they may also be good choices, (Although they carried the New England Idea of Aristocracy, so nothing near jefferson there)

If we are looking for parties, John Adams and/or BEnjamin Lincoln might be popular enough to win for the federalists. Meanwhile, The DRs could elect George Clintoln, Samuel Adams, and multiple others. If you can avoid his death, another federalist may be John Hancock.