How do I move this light box over a few inches in the wall?
The cable comes in from below, and it appears to be running up the right side of the stud, so I'd bet you can swing it over by releasing some staples below the box.
Since you're overlaying paneling or boards, just cut out a couple horizontal slots in the drywall--one at the box level and one a couple feet lower. These could be 4&quot; high by the width of the stud bay.
Now find and pull the staples. Swing the box over and re-mount it to a new framing block, which could be horizontal between studs.
Whether you replace the drywall cutouts is up to you. It would be easy to float some scrap plywood or 1x2 in there and screw them back into place.