Help with electronic controller

[Herpgal]

Please bear with me. I don't even know where to post my question. My husband, who died in 2007, was always working on little projects. I have a reptile room. He built something that enabled him to connect a programmable thermostat to a hot oil heater. Just like a house, I just had to set the day and night temps, and it would turn the heater on and off as necessary.
Well, it's worked for almost 20 years. I did have to replace the little fan on the box, but that was all. Til now.
It's getting power from the outlet into the box, but it's not providing power from the cord that runs from the box to the heater. Well, It has run sporadically. However I am nervous about trying to use it when it has issues, there's a part in there that gets really hot and I can't take chances.
I have tried to find someone in the Syracuse NY area to fix it, but have been unsuccessful, so I thought I'd see if I could find anyone here to try and help me out.
I am probably on the wrong section, so if someone could guide me to the right one, I can repost my plea for help there. The photos show the inside of the box and the thermostat/box set up. The cord on the bottom is the one that is not working properly.
I appreciate any help I getting this running again. If I could hire someone to build a new one, I would. I just don't know where to go. Thank you so very much.

 

[Diver300]

I think someone here should be able to help.

Some more detailed photos would be a big help.

There seems to be a transformer and a solid-state relay. I can't see what the printed circuit board does.

 

[alec_t]

The electronics seem to include a potentiometer, plus an IC in a socket. I'm guessing the purpose is for power control, either by phase-control or discrete cycle control. If that is an IC, it would help to know what's printed on its top.