This is in the archives in more detail, but basically what usually fails
is the fasteners between the armrest "hub" (on which it mounts) and the
seat frame. These have to be tightened or replaced. Use thread locker when
you do.

The armrest is attached to the hub with a pin that slides through two
holes in the metal bushing in the armrest and the middle of the hub
itself. If, from the back of the seat, you push the armrest upholstery
outward, you'll see the head of the pin pointing down (about 4 o'clock if
the top of the vehicle was noon). Make sure both ends of the pin are
visible (so as no to damage the upholstery), then use a punch to push the
pin up and toward the front of the armrest. I believe the pin is beveled
so that only one end should go back in when you've tightened everything
up.

Also see Bentley 72.6.

- Kevin