Peewee King's beautiful song The Tennessee Waltz, although perhaps not strictly a folk song, works well on the uke. It can be done with very simple chords, but sounds nice with these ones:

|G |   |   |C |G |E |A |D |
|G |   |   |C |G |D |G |   ||
|G |B7 |C |G |G |E |A |D |
|G |   |   |C |G |D |G |   ||

E or A could be played as sevenths, as could the D.