What is the best way to light a Lexan (Polycarbonate) sheet?
Cool question.  I have to dig back into my military (Air Force) experience to answer this one. We often used home made light boards to display equipment status and operational sortie info. We built a wooden frame that held the plexi apx 2 to 3 inches away from the wall with a black background panel. We then mounted 8 ft single florescent tube fixtures directly over the plexi, centered on the edge of the sheet, then covered the tube and fixture with a shadow board. This worked great. It illuminated the writing on the board, but blocked the light from coming into the room, creating the effect of neon light writing.  We used colored grease pencils. Red, yellow, green and orange really Pops out!!! I'm sure you can find LED strips that would do the same thing, perhaps a lot more expensive than florescent however. Today I bet you can use a polish, wax or treatment on the plexi that will allow you to use dry erase markers in place of the old fashion grease pencils. Sounds like a fun project, good luck.