According to Wikipedia, 'Stagger Lee' or 'Stack Lee' of 'Stag Lee' was Lee Shelton, an underworld character who in 1895 murdered a rival called Billy Lyons. The original song (probably written in 1897) was a kind of folk-song about this event.
But a quick survey of bands and individuals performing it on YouTube shows that the confusion is even greater than I thought. I found 12-bar blues versions but with differing tunes. One version sounds the same as Frankie and Johnny. And there was an eight-bar version (played by Johnny Wiggs) with yet another quite different tune, sounding suspiciously like the final eight bars of Make Me a Pallet on Your Floor:
Of all the variants, my favourite is the version played by Sidney Bechet and actually entitled Old Stack o' Lee Blues. It's a 12-bar, with a pleasant, simple melody and some interesting chord changes, including what sounds like a tonic diminished on the eighth bar. (It happens to sound very simiar to Faraway Blues.) Have a listen: