Doors – How to fix a door with a damaged knob hole

i used a piece of plywood clamped to the door as an outer-guide. Cut a hole in the plywood, the right size (2 1/8) and then clamp it where you want the new hole in the door. Then using it as a guide on the outside of the hole saw, drill on through the door. voila. no additional tools required beyond the 2 1/8 hole saw, and well, a c-clamp which you should have.

You might want to investigate rim locks. These are decorative/antique styles, but there may be more utilitarian ones if that is what you want.