In this video (marker 4:23), what button is the pilot thumbing on the yoke?
It's the elevator trim control to relieve control forces in pitch. It does this by changing the neutral position of the elevators so that the pilot flying does not need to maintain a constant pressure on the yoke to maintain pitch. On the 747 family, this is achieved by moving the stabiliser.
Although this image shows a home cockpit yoke, the representation of the controls is accurate: