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Is there a technical reason why this wouldn't work?
If it would work it'd produce a light that's energy free and easy on the eyes.
I figured I'd ask about the technicalities before looking for a gb pocket to mangle only to find out there's something fundamentally wrong with the idea.
I think the light it would produce wouldnt be strong enough and also wont last for long.
Phosphorescence needs to absorb photons before releasing them. Basically, you'll need a source of light to charge up the material at regular intervals, which sort of defeats the purpose of your idea.
Charge it in the sun during the day, play games at night...
Why not 
I'm not sure if it'll be strong enough, and if it'll last long enough, but I used some glow in the dark spray paint a while back and I was amazed with the amount of light that came out..
I think you should at least do it for the heck of it.
i tried this some years ago and the results were not very good at all. i was looking at it for a sort of 'passive' backlight approach
perhaps the material has improved and a sheet can generate enough light to be useful, not to mention stay illuminated for more than ~30 minutes, but at the time it just didn't quite work well enough to worry with.
Now that's interesting. I wonder how good are this sheets at light distribution?
I think it might be possible to create a hybrid backlight. Put 2-3 LEDs as in traditional backlights and let them be on for certain amount of time (should be determined experimentally). The ON time should be enough to charge the phosphorescent sheet. And then OFF time should be about the 'discharge' time.
The only point of doing this is battery life and I'm not sure how much it will save 
Still I'd love to see how it looks
I was just thinking about leaving it where there was sun, otherwise using a UV lamp to charge it.
kitsch:
Did you take any pictures of it in action?
looks like someone even patented the idea of a Liquid crystal display assembly with phosphorescent backlighting back in 1986.
http://www.google.com/patents/US4641925