But modifying a DS1287 (soldering an external lithium cell to it) is well-documented and not too hard, so probably worth a try before trashing the whole box.
A new compatible replacement is $10 USD or so: http://www.digikey.com/product-search/en?keywords=DS12887AIn many cases, disassembling, cleaning and reassembling can help. You may end up with different behavior which points to the cause of the problem. Try the serial console, for instance, while leaving the video card out. Try one bank of memory instead of two. Those are hard to find machines!
John