I have a memory of either reading, or being told by a teacher, that the requirement in the Constitution that to be eligible for the office of President, the candidate had to be either a Natural Born Citizen or a Citizen at the time of radification was a political move by Alexander Hamilton's rivals. Because he was born in the Caribbean, he would be ineligible.
A cursory search of the subreddit to see if this question had been asked before revealed several answers to similar questions that maintained that Hamilton was in fact a citizen before ratification and therefore he could run for President if he wanted to.
Did his rivals intend to make him ineligible but crafty wording rendered their intent void? Or did the clause have nothing to do with Hamilton and his rivals at all?
Hamilton was eligible, essentially anyone living in the United States for at least five years at the time of ratification would count as a citizen under Article II Section 1. The clause had nothing to do with Hamilton. He was one of the leading voices to pass the Constitution, and the process of drafting and ratifying the document came before his differences with rivals became irreconcilable. He actually worked with a future rival, James Madison, on the Federalist Papers in order to build support for ratification.
Hamilton remained a viable presidential candidate until the Reynolds Affair and his resignation as Secretary of the Treasury. He also became less viable by publicly feuding with Adams, which would run the risk of his candidacy not drawing the support it needed due to intra-party squabbles.