Um, I guess I'm having a similar problem. I've actually had my FC Mobile II for a few years now, and it had been fine up until today. In my case, the screen is mostly black, though it will occasionally flicker into white. The audio still plays fine, and it also still outputs a perfectly fine A/V signal to the TV with the included cord. I'm actually OK with writing it off, which is why I decided to open it up and see if anything had gone awry, like a wire coming free from the solder or something equally simple. Sadly, there was no obviously fixable problem.
I wouldn't mind getting another portable NES player, but at this point, the FC Mobile II's LCD screen is really really sub-par compared to pretty much every other device out there. It would be nice to see Hyperkin or someone else do a refresh on the portable NES idea because it is really overdue.