Gas boiler pilot light

I have an older Burnham gas boiler. When I set the thermostat at its lowest setting, the boiler will fire until the thermostat no longer calls for heat, then the pilot goes out. Yesterday, I set the thermostat to 60 or so, and the pilot was still on this morning and the system was still hot, so it appears to be cycling. Anyone have any thoughts?

replace the copper wire next to the pilot it's called a thermocouple you can buy one at the big box stores