Actually modern locksets have a 2-1/8" diameter hole, not 2-3/8". The distance from the edge of the door to the center of the hole is either 2-3/8" or 2-3/4".

To drill a hole bigger than the 1-7/8" you already have is fairly simple; you need a 2-1/8" drill bit, which any hardware store would probably have. They're relatively cheap (about $12) and are made specifically for drilling a hole for a doorknob. They're usually in the doorknob section of the store.

Since you already have a hole in the door, drill a 2-1/8" hole into a block of wood or plywood, then clamp that block of wood to the door in the proper position to use as a template for proper alignment when you drill through your template into the door.

nice idea

some search engine work will reveal sources like restoration hardware, antique hardware, etc.

Also retro fit parts for modern sets can be found - although one usually needs to source this info via trade suppliers - often times your local locksmith can source or supply the needed items (a few additional parts from the manufacturer to retro-fit full lockset/knob kits - at minor cost).

You won't find this stuff easily or generally stocked in your local big-box or hardware store - it might take some investigation.

It would he truly sad if you started drilling into old doors in an older home to fit modern (and likely not matching the other) cheapo knob/lock sets.