How do I drill a 2⅛ inch door knob hole over an existing 1½ inch hole?
Two other options. If you need to expand the hole with the exact same center, you can get a hole saw kit that uses an arbor with multiple attachable hole saws. Place the larger size hole saw with the smaller size inside of it, and tighten the arbor on both. The smaller hole saw acts as a guide in your existing hole.



For this to work the smaller hole saw must be the same or longer height, as depicted in the picture above. If they are the same length, the thickness of the base of the larger hole saw will cause it to protrude slightly. You can also use a washer to give it more depth as long as you can still secure the bit to the arbor.

The second option is similar to Chris's suggestion. Clamp a thin piece of plywood over the hole on the door, and drill through that. Pay special attention to keeping the drill level. The plywood keeps you from sliding all over the door. Just make sure you clamp it on well.