Generally, when people make threads along the lines of "What's the earliest so-and-so can be elected President?", someone will post something along the lines of "Well, they reached 35 in year 19XX, so that's the earliest possible, but realistically the earliest would probably be..." However, with some figures, it actually seems that the earliest possible age was not implausible. For example, Henry Clay.
Henry Clay turned 35 on April 12, 1812. At that point, he had already served as Senator from Kentucky for more than a year, served in the Kentucky State House for several terms, rising to Speaker, and had been elected Speaker of the United States House of Representatives in his first term, an unmatched political feat. As Speaker, he was already transforming the position into one of great authority, in part to promote the war against the United Kingdom. From his powerful position, gaining the support of the nominating caucus seems possible.
So, your challenge is to describe exactly how Henry Clay manages to win the election of 1812, either in the Electoral College, or if it ends up deadlocked, in the US House of Representatives. Something will no doubt have to be done to prevent Madison, or whoever is President, from running for reelection, and to block Monroe, the natural successor to Madison, from overpowering Clay and winning the nomination and general election. The main rule for the challenge is that Clay has to be elected to the Presidency, and not succeed just because everyone above him on the line of succession is dead.