How can I fix this old doorknob (circa 1898)?

That lock looks like it's seen some action! How is it still hanging on?

As a basic fix you could get some black phosphate drywall screws (or any black screws to go with the lock, long enough to get well into the door) then drill out the old screw holes - you're not trying to make the holes bigger just deep enough to make a tight fit for a plastic plug (wall anchor) so the screws can get a good grip. The plugs come in different sizes so use the smallest that will fit tight, if that has to be a large plug because the holes are so big already you can put a smaller one or shards of plastic inside it so the screw can still get a good grip in a plug that might otherwise be too big for that size of screw. You might have to cut the plugs down a bit in length which is ok but more length - around 2/3-3/4 of the door width is good if possible, but make sure the hole it tightly packed with plastic. be careful not to drill through the door or into any other screws from the other side. Timber plugs can also be used but this is an easy method that works well as long as it's done right.

These are the kinds of things i mean: http://www.toolstation.com/shop/Screws+Fixings/Wal...

Please DIY safely!