Any part more complicated than a screw or rivet likely has a serial number / part number on it, and if it does, could be traced back to the exact aircraft it was installed on.

And even if there are no serial numbers, there are detailed maintenance records for the aircraft; you can see if a bolt has been replaced, trimmer adjusted or a hole fixed, and you'll find matching records in the books.

This used to be kept in paper log books. Now these records are computerized.

Yes of course, "in the books" was just an idiom here (perhaps not the exactly correct one). What has been done to the plane is a matter of record and can be checked.

Of course. I wasn't disagreeing with you.

Yes. On the aircraft I work on, serial numbers often contain a few digits that are aircraft specific.

In addition to the answers given, I'd have to imagine that the number of 777 airframes that are unaccounted for has to be fairly small.

There is just one 777 airframe that is unaccounted for.

There have been 5 777 hull loss instances (all since 2008) Two crashed just before the runway (Heathrow and San Francisco) one was destroyed by fire while sitting at the gate (Cairo) one was shot down over Ukraine and one disappeared somewhere over the Indian ocean.

Well, to be pedantic, we certainly haven't accounted for all the debris from the MH17 downing in Ukraine. Same type of aircraft, same operator.

But that's very very farfetched.

Zero to be precise. That was the only one missing.

That was the only complete airframe missing, but couldn't there be lost repair parts?

Container ships lose something like 10000 containers at sea annually. Some of those break open and spill their contents. I'd guess that most major airlines stock repair parts in their maintenance centers, and so such parts are shipped across the oceans, and so could get lost at sea.

If, say, Air New Zealand ordered some 777 flaperons to stock their maintenance center, and those got lost in the Indian Ocean, would Air New Zealand just file an insurance claim and order more flaperons from Boeing, and everyone would generally forget about the lost flaperons drifting about in the area?

Or is there some registry or database or some such that tracks these things so that people looking for lost aircraft can check it and find what kind of lost parts they need to be careful to watch out for so that those parts won't mislead their search?

All maintenance on aircraft is closely tracked. They can probably find evidence of maintenance operations that were conducted on the found piece and confirm that it matches the maintenance records for the aircraft. It's a similar idea to identifying people by dental records.

Every part in a modern airplane has a number. Even the smallest piece. These numbers are recorded and tracked during maintenance.

The idea that "everything has a part number" is generally correct but has a few exceptions. You won't find serial numbers (or often, part numbers) on small consumables, like washers, or things like rivets. But it's very common for bolts to have part numbers.

Something as large as a flaperon would definitely have a part number (which would tell you if it was part of a larger assembly), and might be serialized and directly trackable.

Disclosure: I don't work on commercial aircraft, but I do work on military aircraft.