A very thorough inspection with magnification is always a good practice, especially if any sort of corrosion damage is a possibility.
Check the unit's current draw with a fresh 9V battery, with the switch off and the switch on. Should be zero when the switch is off. I can't remember exactly what normal draw is for a 79-III, but it is probably in the 1 to 2mA ballpark.
If the current draw is too low, it could be battery connections, switch contacts, or some of the power supply circuitry has failed open. If the current draw is about normal, the processor and all could be running normally and just the display has failed. If the current draw is excessive, then some of the power supply circuitry has failed short or one of the chips is bad.