"When Archie went to Louisville (x3), Not thinking that he would be killed." "When Gordon made his first shot, O'er behind the bed Arch did drop." "Hush now Guv'nor, don't you cry, You know your son Arch has to die."
This may be based on a historical incident, but there is so little detail left in the song that there is little hope of recovering it; it is hardly possible to look up every Governor Brown in American history.
The final stanza, "Now you see what a sporting life has done, It has killed Guv'nor Brown's only son," gives a clue to what is going on: Archie Brown presumably seduced Gordon's wife/sister/girlfriend/X (somehow the song makes me think of homosexuality, though I can't even guess why), and Gordon killed him in revenge.
This piece is item dF61 in Laws's Appendix II. - RBW