I'm answering my own question here because this took me a while to figure out and maybe others will find the answer useful.

We have a water softener. There are sites out there that explain this in great detail but in a nutshell, there are beads in the water softener that take the hardness out of the water flowing through the tank, and these beads need to be recharged.

The way most water softeners work, every few days (sometimes based on a schedule and sometimes based on a flow meter) at some configured time (2am in our case), the water softener needs to recharge those beads.

This involves flushing the beads with salt water, so the softener runs a good amount of water through the tank that contains salt, through the beads, and then down the drain. This flushes a lot of water down the drain.

Our drain wasn't blocked, but it was slow. I could run water in the laundry tub from the tap for about 30 seconds before it would back up into the tub, and then when I shut the tap off it would very slowly drain out.

So the water softener recharge was putting enough water down the partially blocked drain to dump water on the basement floor, at 2am, every few days.