A few weeks ago our refrigerator starting clicking periodically. Click, then 20 seconds or so and another click. Mostly silent between. A couple of days ago we noticed it wasn't cooling well. We woke up yesterday and found that it wasn't cooling at all.
We moved all the food worth saving out and I pulled the fridge out from it's cubby. I unscrewed the back plate ("Do not remove unless technician"; I guess I'm a technician now) and noticed that the compressor was really hot and nothing else was happening back there.
I did a little research and was led to believe that it was likely the start relay. The modern solid state start relay is a thermistor (thermal resistor) which has fast switching speeds and no moving parts to wear out, compared to electromechanical relays. In theory, this is supposed to make it last longer, in practice, these things burn out (I don't know if they burn out faster than electromechanical relays).
The OEM part number is CLS30820101; the least expensive one I found was at Repair Clinic but they were out of stock. I took a gamble and dropped by a small used appliance shop (their website looks really clean, compared to the store—you should totally go visit them sometime).
I took my broken start relay (with overload assembly) and explained what was going on. He brought out a universal relay (packaging pictured above with broken OEM relay) and explained how it worked. $40 (cash—which I didn't have…I had to find an ATM) seemed like an inexpensive experiment compared to a new fridge.
At any rate, I took it home and in about 20 minutes and a small amount of wiring, it worked: the fridge is cooling and ice is solidifying (which is a side benefit; the ice maker on this model was reliable only for the first couple of years, but the thaw seemed to have cleared something out).
I'm delighted to have spent only $40; I did a little research and discovered that the part retails for $5.95, but the guy at Best Buy Appliance kindly explained how to translate the old assembly to the new one. I figure that was worth the extra money.