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.