Thus, my laptop went through the usual BIOS bootup, and then put up a text screen saying approximately, "Your docking station has no power. Press F1 to shut down."
So, I pushed the power button to turn off the computer, plugged in the power cable, and tried again. This time, I didn't even get a BIOS bootup -- just a character-screen cursor in the top left, and a locked-up computer. A few repeats of disconnecting the laptop from the docking station showed that something was definitely weird-wrong: Outside of the docking station, the computer would boot fine. Connected to the docking station, it would lock up.
The solution turned out to be simple, in one of the sorts of ways that only makes sense in a very twisted sort of logic: I unplugged the docking station, connected the laptop to it, and booted it -- which got me back to the "Press F1 to shut down" error screen, at which point I pressed F1 rather than using the power button. Once I'd done that, the docking station cleared out whatever unfinished quest was confusing it, and when I plugged the power cable back in and rebooted the laptop, things worked just fine.
Crossposted from Dreamwidth (original here), with comments. Comment here or there.