Pseu
The best way to research the history of a song is to amass as much as you can that is extant about the song, including every version available. Then there are several possible approaches; chronological is the most obvious, putting the versions in order that they appeared, were printed or collected, comparing different versions in as much detail as necessary, plotting charts of stanza occurrence is one approach. Another method would be to look at the geographical migration of the song, where it was/is found. Are there any analogues in other languages?

The most difficult ones to trace are those laments that have been on the go for centuries and have picked up commonplaces from all sorts of other songs. The Died for Love family for instance. Whilst these are very challenging they are also very interesting.

A word of advice would be to check if someone else has already done a study on the song you have in mind. No point in reinventing the wheel although you might turn up something fresh or have a different angle on it.