This song was invented by a teenage girl to amuse her little brother whose nickname was Fufu, in the late 60's in Pittsburgh Pa. Several years later this boy was minding some children during a church function, and taught them the song. From there it seemed to spread like wildfire.

The original song was only the first four lines, sung to the tune of "Down by the Station". I do not know who developed the rest of the story regarding the involvement of a fairy. I would very much like to know. Perhaps one day, someone will chance upon this thread as I have, who can answer this mysterious question.

These responses are delightful, the OED2 especially. There was also a camp song that predated bunny Fufu, about a hopping rabbit who was rescued from a farmer, that may cause confusion.

This is truly a modern day folk tale, and I am sure we can all feel free to use and adapt it as we please, thereby perpetuating the story of Fufu into the future!