I think everyone knows "You Are My Sunshine," especially after the release of the movie "O Brother, Where Art Thou?" where it played an important part.
Unlike some state songs, "Sunshine" is easily learned, easy to sing, and easy to play on almost any instrument. In spite of its rather mournful lyrics, it's still a popular tune to teach children in music classes. We sang it in my classes, although I heard my dad sing it before I ever started to school.