In "An Occurrence at Owl Creek Bridge," why does Peyton feel that the idea of hanging is ludicrous?

Let's be very careful here: if you have a look at the text, you will see that this quote you are referring to does not just refer to the idea of being hung. This section of this great short story comes in Part III, when he has been pushed off the bridge, the rope has broken and he has hit the water. Note what the text says:

There was no additional strangulation; the noose about his neck was already suffocating him and kept the water from his lungs. To die of hanging at the bottom of a river!--the ideas seemed to him lucidrous.

Thus what Peyton Farquhar finds ridiculous is the idea of still dying from being hung whilst at the bottom of the river. If you fall into a river and die, you should die of being drowned, not of being hung! We can see how the psychological stress of what is happening to Peyton is reflected in this somewhat bizarre comment. Most people, we would suppose, would have other things on their mind rather than the ridiculous nature of their death.