As far as I've found, most people call it "The moon was shining silver bright." It was apparently written by J. Greener

This is a neat historical song about the presidential election of 1844. Lots of references some people may not get, so I'll copy an explanation of it here:

That "same old Coon" was Henry Clay, Whig candidate for president in 1844.

The "Lokies" are the Loco Focos, the radical Democrats.

The six horses in the pasture are the presumed Democratic candidates; this song was evidently written before the Democratic nominating convention in Baltimore: Polk isn't mentioned (and, indeed, on the day the convention started no one dreamed Polk would come out the eventual nominee).

The "wagon horse from Pennsylvania" is James Buchanan, who had served as both congressman and senator from that state. "The Dutchman" is probably Van Buren (whose first language was Dutch; the only US president who was not a native English speaker). Buchanan is being slandered as a Federalist, and having his statement that ten cents a day was a good wage for a working man pinned to him. Ten cents in 1844 would be about $2.50 in 2016. A wagon horse is not a race horse, and has no hope in a race.

The "old and broken down war-horse" was Richard M. Johnson. Johnson would have been 64 at the time of the election; his fame dates to the War of 1812 when he killed Shawnee chief Tecumseh at the Battle of the Thames. "Rumpsey dumpsey, rumpsey dumpsey, Colonel Johnson killed Tecumpsey" was his slogan in the election of 1836.

General Lewis Cass was, at that time, the US minister to France. In the actual convention he was next-to-last man standing.

The "fiery Southern horse," John C. Calhoun was, at that time, John Tyler's Secretary of State. He'd been Jackson's vice president (before resigning to become Senator from South Carolina during the Nullification Crisis). Calhoun was a fervent supporter of States Rights. Also of slavery. "Matty" is Martin Van Buren, who is seen as the Democratic puppet-master. To "toe the scratch" is to come up to the starting line.

Which brings us to Matty himself, the favorite going in to the convention. Van Buren had run for president and lost in 1840, and run for president and won in 1836. He's "little" because he was physically short.   He also had a reputation for political trickery; hence his symbolic animal was the fox.

John Tyler had been elected vice-president as a Whig. As soon as president William Henry Harrison died (after one month in office), Tyler started vetoing Whig legislation (mostly concerning the central bank of the US). Tyler was thrown out of the Whig party; here the songwriter figures Tyler would run as a Democrat (he didn't).   The "Grippe" was the "Tyler Grippe," the flu epidemic that swept the USA in 1841. Tyler has the distinction of being the only US president who has a disease named after him. The Tyler Grippe's most famous victim was President W. H. Harrison himself. Of interest is that the writer rhymes "Tyler" with "boiler."

Source: https://madhousemanor.wordpress.com/2016/06/06/forward-looking-statements/