I recently fixed a friend's macbook that had this issue - the backlight shutting off as you open the screen. In bright light/sunlight you can see the display is still on and everything else is working fine. When you close the lid, the backlight would come on just an inch before it was closed.
The backlight power cable runs on the left side, it's quite short, and I assume due to poor assembly (refurb?), the cable is easily routed under the left hinge (which holds the cable down) in a way that it pulls it loose in some positions. You may see the connection to the logic board on your left, it's rectangular and tiny.
I removed the display bezel, left hinge and round cover over the center on the bottom of the display. I probably took apart more than I needed to. Reposition the cable so it has a little more slack behind the hinge. You could move it around until it doesn't pull tight and then tighten the hinge down, (and the display backlight stays on in any position).
As far as I know, the problem has not reoccurred. Good luck.