Ever hit your "funny bone"? It's not funny. It hurts.
And it's not a bone either.
What is known as the funny bone is actually a nerve - the ulnar nerve - that runs from your neck all the way to the hand. Commonly, people will whack their elbow on something and the pain will radiate along that path right down to your pinkie and ring finger.
Our nerves are shielded in most cases by bones, muscles and/or ligaments. That is true of the ulnar too EXCEPT at the point where it passes the elbow through a channel called the cubital tunnel. The only protection there is skin and fat, so it is quite vulnerable.
A decent bump there on the corner of some furniture hits the nerve against bone and sends some pain down the forearm and hand.
So, why call it a "funny bone"?
There seems to be a two-part answer. The ulnar nerve runs along a bone called the humerus (homonym for "humorous") and at one time it may have been believed that it was that bone that was reacting.
It might also be that "funny" was used in its non-humorous sense of "odd" as in having a funny feeling about something.