As you may know, in the American political system, the people do not technically vote for the president, instead, there is the Electoral College. To simplify it, each state is assigned a certain number of electors, and the voters choose electors pledged to each candidate. However, an elector can vote for whoever they want, and occasionally there are so-called "faithless electors", who vote for candidates other than the one to whom they are pledged.
This is generally insignificant, as a single electoral vote the wrong way rarely matters, and faithless electors have never changed the outcome of a presidential election. Your challenge is to create a situation where they do.