Margaret suggested to me that the point of one's elbow is called the "funny bone" because it's near the end of the upper-arm bone, the humerus. "Humorous" … "funny bone" …
The pun is thought-provoking as an etymology. I happen to think it's a coincidence: First, most people don't know the Latin name humerus, leaving few who would be eager to propagate the pun. Second, there is a regional American term "crazy bone," meaning "funny bone"; "crazy bone" obviously developed without being based on a pun; if "crazy" could be used without needing a pun, why couldn't "funny?"
If I'm right and Margaret is not, her etymology might be what's called a false etymology. If you think false etymologies are interesting, also look at the Wikipedia article on the distinct phenomenon of folk etymology.
By the way, the scientific term for the funny bone is apparently "olecranon."