What resistor
An interesting protection circuit. page HH-16
Current gets to the compressor "run circuit" via either the blue wire (which draws from the overheat bi-metal) OR via the resistor green wire in series with the "auxillary winding" inside the compressor.
If running normal, the auxillary winding / resistor allows enough bypass current to flow in them to the run winding that the overheat bi-metal doesn't heat up enough to trip.
If the compressor is just starting or is stalled due to brown out or other compressor maladys, then the impeadance in the auxillary winding is high, cutting off current in that path and forcing all "run" current to flow the bi-metal overload and hence to trip the overload if let go too long.
As said on page HH-16, your results will vary, if the resistance is too high, then the overload will trip during normal running. If too low, then it won't trip when it should during a stall.
Play around a bit and try some values. I'd go conservative (higher), but if you have brown outs you may want to choose a lower value.
Being able to measure the current in the green wire and blue wire would be a big help here.
(If the failure is not due to resistor corrosion or fracture, is due to high compressor current draw during normal run, then the resistor may pop again.)