Any thoughts on what this might be?

I've done a bit of checking into other possibilties... Nothing loose under the deck. Nothing loose under the flywheel housing. Flywheel shear pins are intact. The magneto is providing spark on my spark checker, but I don't know what the advance is supposed to be -- what I can say is that when the piston reaches at TDC, the magnets are almost exactly 90 degrees past the magneto. The piston is reaching the top of the block (I've taken the head off to look). The valves are operating at the appropriate times and nothing is broken around the valve springs.

Any further insights?

Then it hit me: as the magnets were going by on the flywheel, they pulled this piece closer. And a close inspection of the flywheel magnets showed tiny scrape marks where this piece has been pulled in close enough to make contact with the spinning flywheel. Smacking the piece with a screwdriver did reproduce a sound similar to what I'd been hearing.

So when I get a chance, the next step will be to put everthing back together and see if bending that metal piece out a bit stops the noise. If not, I think the only thing left to do will be to drain the oil, pull the bottom off the motor, and examine the piston rod to see if maybe it's loose where it attaches to the crankshaft, although at this point I'm not sure how it could be since the piston seems to be operating as it should and there are no odd sounds when I spin the engine by hand.

Thanks!

FWIW, it's somebody else's mower, so I can't take credit for hitting whatever was hit. :-)

Although the flywheel shear pins are intact, I can't help but wonder if the timing isn't getting screwed up somewhere. One other bit of info that makes me suspect timing: every 10 pulls or so, the starter rope kicks back wickedly and goes flying out of my hand, and I keep thinking maybe there is someplace that there's an ignition spark when it shouldn't be.

I'm seeing references on the exploded engine diagram to a "timing key" and "timing gear" on the crankshaft. Could those be the culprit? Or do those have to do with valve timing rather than ignition timing?

The flywheel key was sheared, so the flywheel was about 60 degrees rotated from where it was supposed to be. I think the clanking noise was a combination of the aforementioned metal piece slapping against the flywheel magnets, plus the sound of pinging/knocking resulting from the fuel igniting when the cylinder wasn't ready for it. Once I managed to get the flywheel off and installed a new key, the engine ran fine.

THANK YOU to everyone who chimed in.

'The flywheel key was sheared'

I don't know how it wasn't and then it was, but that's exactly what the key is supposed to do - shear, so the flywheel can shed it's energy without bending the crank. That's also why blades are so ductile, in that they'll bend double without cracking. You're very fortunate that the crankshaft wasn't damaged.