Looking through early American history and found some interesting people:
John Dickinson (1732-1808) - Penman of the Revolution and author of the "Liberty Song." Attended the Continental Congress, the Annapolis Convention, and the Constitutional Convention. However, while he supported the Revolution, he was extremely reluctant to embrace secession from Britain and opposed the Declaration of Independence when it was first proposed. This would make in an interesting choice for president, but would also serve as fodder for his enemies. He was the largest slaveholder in Delaware, but due to Quaker influences in his family, he freed all of his slaves between 1776 and 1786.
William Few (1748-1828) - Famous rags-to-riches frontiersman before Andrew Jackson or Davy Crockett. He was born in Maryland and moved to North Carolina, where his family got involved in the War of the Regulation, a populist revolt against British authorities from 1765-1771. Few joined the Georgian militia during the Revolutionary War and later represented the state in the Constitutional Convention and the first Senate. He supported the Constitution, but joined the Democratic-Republicans after its ratification and opposed Federalist policies like the National Bank. He moved to New York City in 1799 and continued to hold local offices. He comes off as a Cincinnatus type who didn't want to hold public office, but kept getting elected to it anyway. Unlike most Southern Republicans, he opposed slavery.
Gouverneur Morris (1752-1816) - Most famous for writing the Preamble to the Constitution. In politics, Morris was a typical high Federalist like his friend Alexander Hamilton, supporting a strong federal government, lifetime appointments of officials, and a quasi-aristocracy. However, he's probably also the Founding Father with the most fun facts about him: he had a peg leg from a 1780 carriage accident, carried out an affair with Talleyrand's mistress in the Louvre while Ambassador to France, first married when he was fifty-seven to a thirty-five-year-old that had previously been accused of murdering her own illegitimate child, and died after attempting self surgery of a urinary track blockage using a whalebone catheter. During the War of 1812, Morris advocated for New York/New England secession from the Union, which could also make him a potential leader of an independent New England.
John Taylor of Caroline (1753-1824) - Virginian Anti-Federalist politician/philosopher who strongly supported states' rights, agrarian democracy, and secularism and wrote some early critiques of capitalism. However, he was also a proto-Calhoun who provided intellectual defenses for slavery even though he thought it was immoral. Could serve as a stand-in for Thomas Jefferson or other Democratic-Republican leaders.
Albert Gallatin (1761-1849) - Swiss aristocrat who got bored of college and moved to America with a friend to join the Revolution. He led the Democratic-Republicans in the House, served as Jefferson and Madison's Treasury Secretary (during which time he advised them against abolishing the national bank and became an early advocate of internal improvements), kept the nation's finances in order during the War of 1812, and helped negotiate the Treaty of Ghent. The closest he came to the presidency was 1824, when he was chosen as William Crawford's running mate, but he didn't want the position and withdrew due to his opponents' smear campaigns. After retiring from politics, he formed the American Ethnological Society, which studies Native American language and culture. He would be a good choice for a foreign-born president (eligible as he was a citizen when the Constitution was ratified).
EDIT: Another interesting one, though unlikely to become president
Humphrey Marshall (1760-1841) - An "aristocratic lawyer with a sarcastic tongue" most famous for his 1809 duel with Henry Clay. He served as Senator from Kentucky from 1795-1801, but, as a staunch Federalist, failed to win reelection. Additionally, he was controversial for being "zealously antireligious" and self-publishing a number of pamphlets criticizing religion. Comes off as sort of a proto-H.L. Mencken.