It could be a lose connection or cable. Temperature changes cause them to contact and expand. Is it just the screen not working when cold or is it actually not booting? Try it with an external monitor to be sure. It could be the LCD screen is bad. If it is not booting, try boot from a CD. It could be a hard drive.
Laptops are often not economical to repair. A replacement LCD screen can cost as much as new laptop. Hard drives are little less expensive, but still consider if the cost is worth it. For an older laptop I would probably replace it instead of repairing it.