I think the name you have is
The marching tune is referencing the civil war. The name could be Abraham Lincoln... or possibly, the Confederate States of America.
"And a one, and a two, and a three four five six!"
The initial Confederacy was established in the Montgomery Convention in February 1861 by seven states (South Carolina, Mississippi, Alabama, Florida, Georgia, Louisiana, adding Texas in March before Lincoln's inauguration) Texas doesn't count here because it was before the inauguration.
"Or a one, and a two, and a three four five!
-- expanded in May–July 1861 (with Virginia, Arkansas, Tennessee, North Carolina), and was disintegrated in April–May 1865. It was formed by delegations from seven slave states of the Lower South that had proclaimed their secession from the Union. I count Texas here. These eleven states eventually formed the Confederate States of America.
"From my home right here,
To the end way down there!"
Lincoln lived in Springfield, IL. The confederacy went as far south as Florida/Texas.
"And then I have a choice
Of a one, a two, or a three, or a four!"
The Crittenden Compromise was an unsuccessfully proposed to Lincoln, a proposal to permanently enshrine slavery in the United States Constitution, and thereby make it unconstitutional for future congresses to end slavery. The compromise proposed four Congressional resolutions.
"First named him in 1861,"
On March 4th, 1861, Abraham Lincoln was inaugurated (named) as President of the United States.