Either the display, the traces (usually ribbon connectors), the power to the board/display, or the support chip could be bad. I would call Sony support (parts) and tell them you need a new display, and see what they can do. You would pretty much need a schematic to troubleshoot this problem for sure, but sometimes you can reseat a ribbon connector or resolder some cracked traces if you tear into the unit and inspect it carefully.
Sometimes the price of parts/boards deters people from fixing the unit, and just living with it.
Perhaps Sony will be an option, give it a try. Here's their contact site for parts:
http://esupport.sony.com/US/perl/model-accessories.pl
Good luck, and hope this helps.