Refrigerator still clicking after replacing start relay

About 50% of the time, the fan comes on but the compressor doesn’t. After about 10 seconds, there’s a loud click. About 1 minute later, there’s another loud click, and the compressor comes on. The refrigerator cools ok.

I removed the start relay, and I can hear it rattle. I replaced the start relay and the capacitor. The compressor still clicks about 50% of the time.

Should I try yet another start relay? What else could be wrong?

Was the overload replaced at the same time as the start relay? Sometimes the start relay can rattle even if it is good. I sometimes hit the compressor with a rubber mallet at the same time it tries to come on, which will usually go ahead and start, which shows me the start relay is bad.
You need to test your relay after you take it off. Test for continuity. No continuity means bad start relay.
Lastly, the compressor may be getting ready to frag out.

CletusT, thank you. When you say "overload", are you referring to the capacitor attached to the relay? If so, yes, those were replaced at the same time.

I will pull the relay and test for continuity. What should I expect when I test it?

Yes it looks like all one unit on your machine, part AP3997558. You have replaced all components that attach to the compressor it seems. Just needs to show continuity between terminals on the start relay.

Ok, I’m reading 7.9 ohms between two of the terminals on the start relay. Does that sound right?

If this turns out to be a dying compressor, do they normally fail suddenly or will the problem just get progressively worse?

That sounds good, the overload is the little white and brown, or white and black triangular shaped piece. It should have some sort of continuity between the terminals as well.
Compressors fail in two ways, mechanical failure, which includes locking up, or in an electrical fail, which is when one of the windings gets burnt. They can go suddenly or slowly. When they are dying a slow death, they often make a lot of noise.
Once a compressor locks up, or windings burn out, it will sit and click over and over without starting.