A black mirror? Much like a black Google blackle.com , without hard evidence, my guess is the energy savings is almost none.

Since my first phone with a background c. 2010, I have never changed the background from whatever the default is. I keep waiting for something important enough. I should probably reflect on that at this point.

If it's a OLED screen, than this is a pro tip.

Never did it differently.

But with non-OLED, this isn't useful to much. Pull the brightness down. Which also a pro tip for OLEDs :)