In Lincoln's 1860 campaign republicans took an old song called "Down in Alabam" (melody similar to the currently known Old Grey Mare) and reworked it to suit their candidate.   

I gather The orginal had verses like:
"Old grey horse came tearing out the wilderness
Tearing out the Wilderness,tearing out the Wilderness
Old grey horse came tearing out the Wilderness
Down in Alabam"

This was reworked for the campaign as:

"Old Abe Lincoln came tearing out the wilderness
Tearing out the Wilderness,tearing out the Wilderness
Old Abe Lincoln came tearing out the Wilderness
Down in Illinois"


The song - either the orginal or the parody or a combination - subsequently became a play-party song with all sorts of floating verses that fit the basic formula. Some how the "Old Grey Mare she aint what she used to be" verses came in (a clear reference back to the "old grey horse" but possibly a political statement of their own), stuck and took over the title of the song and represents the most commonly known survivor of that family.

Another variant of the song (possibly another verse)is "The Old yellow dog came trotting through the meeting house", which remains the title of a popular southern and midwestern fiddle square dance tune/song, and which has a chorus with a different tune which goes "brave boys die, down in Alabam", which could derive in some way from the original song. I have heard theories on both sides as to whether or not the "Old Yellow Dog" is a political reference that was used in a campaign.

I can't say I have thoroughly researched the issue though so perhaps others have some better information.