The song that the Tinkers are looking for would have to be the ancient song of the Da'shain Aiel. It's said that during the Age of Legends the Ogier, Nym (the Green Man's species), and the Da'shain Aiel (Tinkers/Aiel) used to use songs to aid the growth of plants in the Age of Legends. The song wouldn't have anything to do with the One Power - it would be something similar to the Ogier Treesinging.
Maybe this song could be used to clean the Ways? After all, the Ways are living things and the Ogier "sing" new Waygates. Maybe if the Ogier and Tinkers work together (too bad the last Nym died in the first book) they can cleanse the Ways of Machin Shin.
wotmania says: I think it is a pretty reasonable guess the song the Tinkers are looking for is that song from the Age of Legends. Using it to clean the Ways is an interesting possibility. I hope Jordan goes far enough after Tarmon Gai'don to see things like that resolved. Of course, if he does resolve everything, what would we argue about after the series is over?
Erith 
Shel
That's my bit
Yah I like that...