There are several contractions of words that I cannot figure out in Twain's writing: "That last fight of his'n, and the way it turned out." ...

"His'n" is a slang contracted version of the words "his
one," meaning something that belongs to him. It derives from the dialect of the
19th-century American South. Similar contractions appearing around the same time are
"your'n" and "her'n." The contraction would seem to refer to a singular possession, but
as its usage became more common, it came to be used in reference to singular or plural
possessions.

Usage of the word tends to be by rural populations and
appears to transcend racial boundaries.

I found the word used in the
1853 book "Weren't no good times: personal accounts of slavery in Alabama," edited by
Randall Williams. And Sojourner Truth is quoted as using the word in 1858 in the book "A
scholar's conscience: selected writings of J. Saunders Redding, 1942-1977" by Jay
Saunders Redding and Faith Berry.

In more current day usage, former
Houston Oilers head coach O.A. "Bum" Phillips was quoted in numerous sources in November
1992 as praising legendary Miami Dolphins head coach Don Shula in this way: "He could
take his'n and beat your'n; or take your'n and beat
his'n."

It's no surprise you found the word in a work by
Mark Twain. He was a master of the American Southern
dialect.

Hope this helps. Good
luck!