What is the etymology of seat-of-the-pants?
I believe the full phrase started as flying by the seat of your pants and it comes from the aviation community.
When flying an aircraft a pilot can rely on instruments, e.g., a turn coordinator, to tell him whether he is flying his airplane in coordinated flight or, he can rely on the way it feels. The pilot can feel whether the force keeping him in his seat is directly straight down into his seat as it is when you are sitting in a chair on the ground or whether it is going through the chair at an angle- picture yourself sitting on a chair on the side of a hill. You really can feel the difference in the seat of your pants. As StonyB points out, this article from Popular Mechanics mentions that a pilot can also feel when the aircraft is climbing or diving by sensing how "heavy" or "light" he feels in his seat.
One should also note that the part of the pants or trousers that covers the buttocks is known as the seat. (4b)