The elbow is made up of three bones: the humerus (upper arm bone) and the radius and ulna (forearm bones). At the bottom of the humerus on the inside of the arm is a bump called the medial epicondyle. Some people call that your "funny bone."
The ulnar nerve of the arm passes around the medial epicondyle. There's a slight indentation or groove in the bone where the nerve fits. The muscles and ligaments around this area form a tunnel through which the nerve passes. The tunnel is called the cubital tunnel.
Even though these structures protect the nerve, the groove is very shallow so the nerve is close to the surface. Any time you bump or hit the medial condyle or the nerve underneath it the nerve gets dinged and you get a strange sensation. Some people describe it as painful; others say it's a sharp sensation of pins and needles.
Either way, it's not very funny. Most likely it gets its name as the "funny bone" because someone else finds it funny to watch the person who bumped it jump around in pain.