The name of the piece of hardware is a friction lid support. It screws into the side of the taboret. The slotted part attaches to the base for the palette, which tilts up and down on door hinges. By tightening the crank, you can clamp the palette at any angle.
The wooden bar is for the oil paint that you squeeze out from the tubes. The white palette paper is for mixtures. As mixtures dry up and are no longer needed, the paper tears off the side of the palette.
I got this basic idea from John Berkey when I visited his studio almost thirty years ago. One difference was he had a take up spool with a crank on the right side of the palette.
Previously: Taboret (with more pictures)