The funny bone is not a bone
Bang your ELBOW in the wrong spot and an electric tingle shoots all the way down to your little FINGER. Nothing has been bruised. You have squashed the ulnar nerve, which runs from the neck to the hand and crosses the elbow in a shallow groove with almost nothing on top of it to protect it.
It is the only major nerve in the body that travels that exposed, which is what makes the sensation so strange and so specific. The name is probably a pun on the arm bone it runs across, the humerus, and the fact that it is not remotely funny at the time.
These are the oldest words in the language
Words for body parts barely change. Linguists comparing languages lean on them heavily for exactly that reason: EYE, EAR and NOSE survive for thousands of years, along with words like tooth and blood, while the words for tools, clothes and money come and go.
English EYE traces back through Old English to an ancestor it shares with the Latin oculus, which is still visible in words like ocular and monocle. The same is true of EAR and Latin auris, and of NOSE and nasal. A four-year-old learning to point at their nose is using a word that has been passed down, more or less intact, for several thousand years.
Your body is all over the language
English is stuffed with body parts doing something other than their job. You give someone a HAND, keep an EYE on the baby, are up to your NECK in it, pay through the NOSE, put your FOOT in it and pay an ARM and a LEG for the privilege.
Other languages do the same thing with entirely different parts. Where an English speaker learns something by heart, a French speaker learns it par cœur too, but a Spanish speaker who is fed up is hasta las narices — up to the noses. Body words are the first metaphors any of us learn.
Head, Shoulders, Knees and Toes borrowed its tune
The melody is not original. It is usually sung to the tune of a much older song, There Is a Tavern in the Town, which most of the adults singing it have never knowingly heard.
The song works for the same reason pointing works. Saying HEAD while touching your head attaches the word to something the child can feel, and a word attached to a movement is remembered better than one that is only heard. Finding KNEE or ELBOW in the grid and then wiggling the real one is the same trick in slow motion.
From the top down
The grid runs the whole way: HEAD and HAIR up top, then the EYE, EAR, NOSE, MOUTH, LIPS and CHIN, down the NECK to the ARM, ELBOW, HAND, FINGER and THUMB, past the TUMMY and BACK, and on to the LEG, KNEE, FOOT and TOE.
There is a practical reason this topic comes so early. A child who can say which part hurts gets help faster and gets the right help, and TUMMY in particular does an enormous amount of work in the first few years of a child's life.