I think Belly Drum is based on Taiko, which is a type of a Japanese Drum.
In feudal Japan, taiko were often used to motivate troops, call out orders or announcements, and set a marching pace; marches were usually set to six paces per beat of the drum.
Taiko drums and drumbeats have been a cultural phenomenon that is supposed to rally courage, appease the sprits and so on. They're a fairly common Japanese icon. They were probably used to create fear within opposing enemies and so on.
As for the HP halving, I can only assume KRLW890 is right about the Pokemon literally hitting itself hard enough to cause damage. That, and the devs felt that was a good way of balancing such an otherwise broken move.