In Part III, which mostly takes place in the feverishly-paced imagination of Peyton Farquhar during his fall from the bridge to the end of his literal rope, Farquhar imagines that the "rope [has] broken" and that he has fallen into the river beneath the bridge. Thus, the circle of the noose would be, in his imagination, still pulled tightly around his neck, the rope having snapped higher up. This is why he feels that it "was already suffocating him." The narrator tells us Farquhar's thoughts in this moment, saying,

To die of hanging at the bottom of a river!—the idea seemed to him ludicrous.

Thus, it is not the idea of hanging, or even of drowning, that strikes him as ridiculous, but, rather, it is the idea of dying of hanging while he is not actually hanging but is submerged underwater at the bottom of the river. To die of hanging while one is actually hanging would not be ludicrous, but to die of hanging while one is not actually hanging strikes Farquhar in this moment as absurd. Such a thought at such a time might be read as one of many clues for us that not everything is as it seems.

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.