In theory this should be easy -- have the second thermostat connect the R and G wires when it calls for cooling, and the fan will run.
Here's one reason it might NOT work, and it's subtle/non-intuitive so I thought it was worth mentioning: Some thermostats short the Y (cooling) and G (fan) wires together whenever they're in cooling mode. They do this so that they can use one relay to turn on both fan and A/C during a call for cooling. (Relays are relatively expensive components.) This shorting happens whenever the thermostat is in "cool" mode, whether it's actually calling for cooling or not. (Although if you set such a thermostat's Fan switch to "ON" instead of "AUTO", the short is removed so as to run the fan only.)
Where there's only one thermostat in the system, this shorting behavior does no harm. But if a second thermostat is added and it calls for "fan", the power will go through the short in the first thermostat and turn on the A/C as well. I have this exact arrangement in the house I just bought, and I cannot run the fan independently from either thermostat -- it will always run the A/C as well. If I unwire either thermostat (or set it to "heat" mode), I can then run the fan without the A/C.
So you might try shorting R (or Rc) and G with your existing setup to make sure that the A/C does not come on as well.
Source: personal observation confirmed by isolated testing on Robertshaw 9600 programmable thermostat
After looking at all the photos, it looks like the fan is controlled by a fan and limit switch (the black thing with a silver dial and three black wires).
So basically, the thermostat calls for heat by energizing the TH W terminal on the gas and ignition controller (Black Honeywell box with all the wires connected to it). This opens the pilot, turns on the spark, and ignites the main burners. Once the temperature in the heat exchanger gets up to the FAN ON temp., set on the fan and limit switch, the blower comes on. When the thermostat is satisfied, it removes power from the TH W terminal. This tells the ignition and gas controller to turn off the main burner. When the temperature in the heat exchanger falls below the FAN OFF temperature setting, the blower turns off.
To control the fan via the G wire from the thermostat, you're going to have to use a relay to turn on the blower. You'll energize the coil of the relay using the G wire from the thermostat, and the 24V common terminal from the transformer. You'll have line voltage (120V or 240V) connected to the common terminal on the relay, and a wire to the blower connected to the normally open (NO) terminal on the relay.
If you don't understand the information provided in this answer, or don't feel comfortable making this type of modification to the furnace. Please don't hesitate to contact a local licensed HVAC technician.
Side Note: The gas and ignition controller is installed upside down
Best Answer
Yes thermostats do go bad. The old double mercury thermostats are great for simplicity and no batteries needed.
Yes digital thermostats do fail. Just like all electronics they have a life and if in an electrical noisy area the life will be shortened. I have only seen a few thermostats that were actually integrated as part of the heating / cooling system.
Depending on the features your system has you will need a common wire a call for heat and a call for cooling, fan only, different fan speeds / heating levels . You can install the old thermostat and although it won’t cost anything today loosing some of the features of a modern digital may be worth the expense. Most only have cooling/ heat a fan only setting and off.