Unfortunately, luminescent materials aren't that strong. They wouldn't produce a suitable amount of light to view a screen image in the dark. (especially considering the material would be located behind the screen, reducing the amount of light it might capture)

Your best bet is to use the same kind of principle found in watch backlights:

"In an Indiglo watch, a very thin panel uses high voltage to energize phosphor atoms that produce light. The panel itself is extremely simple. As described in the Timex patent, you take a thin glass or plastic layer, coat it with a clear conductor, coat that with a very thin layer of phosphor, coat the phosphor with a thin plastic and then add another electrode. Essentially, what you have is two conductors (a capacitor) with phosphor in between. When you apply 100 to 200 volts AC (alternating current) to the conductors, the phosphor energizes and begins emitting photons."

Remember that any patented device must provide instructions for how the device is assembled, so the general public can reproduce the device for themselves. You can find detailed instructions for this sort of thing online.

Having four AAA batteris to power it, this kind of lighting would probably work well without diverting much power from the calculator itself. In fact, you could shunt it over to the back-up battery, and probably operate off it for a good two hours.

The only problem with this is that you need AC current to light the phosphor, not DC (which is what batteries produce). You'd have to learn how to make the program that powers this produce a AC current from DC, which I have no clue how hard that would be or not.

As for the glow in the dark tape...I can kind of guarantee that it won't work. The tape itself is really thick so that is the big issue there. As for glow-in-the-dark paper, I've never heard of it but I suppose that could work, in theory.

For the external (I'm assuming that is what was meant) LED, you would need to do all the physics behind it to make sure you have the correct current, power, voltage, and any resistors/capasitors if necessary. Then make the program to execute those exactly haha.

In short, its not easy haha.

agreed.

Unless you get electrical photo luminescent tape   My father used some to hack in a backlight to a LCD display recently and it turned out really well.  IIRC it does require a higher voltage or something tho so it might be a bit of a hassle to install

But it wouldn't be as cool.