What Is The Origin Of The Word "President"?

It was first used at Harvard University.   The term was coined for himself by the man who became the president of Harvard in 1640, Henry Dunster.    A President was one who presided over the various colleges that made up the University.  Soon the term caught on for other young universities in the American colonies.  

Moreover, within the government of the American colonies (under the British) there included a person who "presided" over the local assembly, but this person was not actually called a "president".

Some of the people who co-wrote the American Constitution (including James Madison, Thomas Jefferson and Benjamin Franklin) had received, added to research themselves, or directly supported (through donations) higher education. So they knew about the idea of a 'President" as the name of a person who presided over a university.  The group of colleges was rather like the sometimes fractious group of States trying to unite under the umbrella federal government of the newly independent USA.  

Madison and his contemporaries were also very much men of the Age of Enlightenment, when Reasoned Learning was held in high regard.  So they would have looked to Academics for inspiration in many matters of politics and science.