Ask HN: Does anyone have a good explainer on miner selection in PoS blockchains?

I was talking crypto with my friend yesterday, and I mentioned that in a Proof of Stake system, it's much more energy efficient because only a small randomly selected subset of miners verifies each transaction, so they aren't all solving crypto problems to participate.

He then asked me what the random selection algorithm is that maintains security with possible bad actors, and I didn't know.

I tried looking it up but I'm failing to find the details, everything just says "they are randomly selected with a secure algorithm". Can anyone tell me what the algorithm is? I know (think?) that it's a variation on Byzantine fault tolerance, but I can't find the details.

Thanks!