I would check to see if the pot that controls the contrast is functional with a multimeter. If it isn't, replace it and see if that solves the issue.
I would also check all components on the PCBs for signs of damage, e.g. burnt transistors, charred board, leaky/dry electrolytic caps... If you identify any bad components, solder in a replacement and see if that works. There are some decent Game Boy schematics on dev'rs if you need. Link: http://www.devrs.com/gb/hardware.php#har...
From there, there's not a lot of places to go. The most likely cause of the issue is what Andrew said. The thin ribbon cables used to drive the LCD were actually sandwiched inside the panes of glass that constitute the LCD. There is a thin rubber strip with adhesive running along the bottom edge of the LCD to create pressure when the case is closed in order to maintain a solid connection.
Over time people would frequently (even if inadvertantly) expose their Game Boys to sun or other heat sources and these ribbon cable connections, particularly the bottom one, would come apart from the LCD. This is what commonly causes lines in the LCD or complete lack of LCD function.
The fix for this problem would be a replacement LCD screen, but finding a working Original Game Boy LCD screen by itself is fairly unlikely. Your best bet would be to buy a known working Game Boy and keep this one as a spare parts source for the working one.
Most likely you have a problem with the connections to the screen. They are fairly fragile and over time can malfunction and be completely un-usable. At this point you can go inside the game boy and look to see if anything has come apart, but your best bet would be to find a new Game Boy.