The origins of this can be traced back to CINA (the Commission Internationale de Navigation Aerienne / the Convention for the Regulation of Air Navigation) established as part of the Paris Peace Conference immediately after WWI.
A part of this conference was the adoption of a system of international aircraft identification still in use today. The U.S. delegation was allocated "N".
A number of theories exist as to why "N" was chosen specifically, and there's a great article about it over at the American Aviation Historical Society.
Theories range from, the largest number of states begin with the letter N; to the recognition of the development of wireless communications by the U.S. Navy that had been using "N" as the prefix to its station call-sign identifiers since 1909.