What is this button in the cockpit for?
It's taken a while, but I've finally worked it out... It enables / disables the trim control on the yoke (which you have hidden in the photo).
The way to spot it is to go to the X-Plane General Settings dialog and enable the "Show clickable regions in the cockpit". Anything you can click has a green box drawn on it.
When the button you're asking about is "On" (in), the pitch trim control on the yoke shows manipulators on it; when its "Off" (out), the pitch trim control has no active manipulators displayed.