chew the fat

Etymology
One of the possible origins is from the fact that after a meal people would go into their sitting room with their guests and chit chat while chewing on the fat left over from a meal.

Another possibility comes from sailors who often had salted beef and pork on long voyages and the fat would harden. Sailors would eat and complain about the hard life of the sea while "chewing the fat"

Probably just from comparison of the jaw movements in chewing to the jaw movements in talking.

Verb

 * 1)  To chat idly or generally waste time talking.