Fast forward to mid-afternoon, when I go back down to see how things are doing. The puddles are back and they're bigger. And it's a gorgeous sunny day outside, so it definitely ain't the rain. It was then that I noticed that the water seemed to be coming from the vicinity of the furnace. Back when I built the prototype for my bookcases, I wound up placing it directly in front of the furnace, which means I don't really see the furnace all that often anymore. When I looked back there, I noticed water dripping from the pipe leading from the pressure relief valve. Tried fiddling with the valve to see if I could get it to close properly, and the drip turned into a stream. Just. Frelling. Great.
Shut off the furnace, and the water going to the furnace, then I got out a big wrench, and removed the valve from the system. The gasket was frelled, and a loose piece of it was jamming the valve open. Put it in my pocket, and headed out to the Depot, who it turned out didn't have what I needed. Bother. They had one valve that could've worked, but it had a female fitting instead of male, so I would've had to add an adapter, plus it seemed snazzier and pricier than what I actually needed. Went over to Mancini's to see if they had one, but they were already closed. Tried a couple of other places, but again no luck. So I went home, and reinstalled the old valve until I can track down a new one. Removing the old gasket helped. It doesn't seal completely, but now there's only a drop every minute or so imstead of a constant stream. I put a bucket under it to catch the drops, then I vacuumed up the remaining puddles. Now the floor is finally drying out. Tomorrow I'll see if Mancini's has the right valve.
Two things - First, I should've trusted my instinct about the weirdness of the water at that end of the house. Second, I am very happy that it was a malfunctioning valve rather than some nebulous leak from the outside. The valve I can fix.
Federal taxes are done, and I'll do the state taxes tomorrow. Getting a decent refund, so that's good. New windows for the house.