What does by the great horn spoon mean?

It's a very old-fashioned American oath. A horn spoon is just a spoon made out of horn, like cow's horn - people used to carve their own spoons back before mass industry made cheap metal eating utensils. There have been hints that the saying originally referred to the Big Dipper, and was used by sailors.

The first record of this saying was in a song from 1842 called "French Claim" -
The more he thought on't it the madder he grew,
Until he vowed by the great horn spoon,
Unless they did the thing that was right,
He'd give them a licking, and that pretty soon. It doesn't really mean anything other than "I swear" - but you know how Americans love using five or six Words when one will do just fine!