I spent 9 years selling and trouble shooting heating and air-conditioning equipment, but I’ve been out of the business for 20 years except for repairing my own stuff as it needs it. So I do have some expert knowledge in this area, but it’s a little old. Take that for what you will.
The price quoted seems high to me. I would have guessed more like $500.
What you’re describing is called a “direct spark ignition” system. When the thermostat calls for heat, several things have to happen in sequence to safely light the main burner, keep it running for the correct amount of time, and shut it down.
I’m guessing by “brain” the repairman is talking about the electronic board that controls this sequence, and has all the controls and sensors connected to it. For a residential furnace, this will be the spark unit, flame sensor, high-temperature cut off, combustion air?flow switch, gas valve, any safety interlocks, the thermostat, and possibly relays to control the main blower motor.
Now bear in mind that replacing these boards, or anything else in a gas heating system IS NOT SOMETHING YOU DO YOURSELF unless you’re not only comfortable with basic electricity and control systems, but also familiar with gas heating systems. For most people, this means you need a repairman.
The last time I bought a control board (for my own furnace, five or six years ago) it ran about $150. In my day, it was common for HVAC repair companies to mark up their parts about 100% (i.e., double the price). By the way, by googling for you brand of furnace and “control board” you may find some hits on the board you need. Be careful, there is more than one kind of control board.
If the part is still available, any competent repairman should be able to troubleshoot the problem, and replace the part in an hour or two (not counting the time to chase parts). Another hour or so to test the unit through a few cycles, check for other potential problems, and that should do it.
So, assuming the board runs $150 to $200, and they charge $75 per hour, your cost would be $300 to $400 for the part, and $150 to $225 for labor.
You know how long it took him to troubleshoot it, and it shouldn’t be any more than another 2 hours to complete the repairs, unless there is some other problem that can’t be found until this part is working (possible, but unusual).
I’d say get a second estimate, and make sure it’s itemized for specific parts and labor costs.