Large Hole in an oak wood cabinet drawer

Where in the drawer is the hole? Exactly how large is the hole? A picture would be invaluable.

What tools do you have to work with?

You'll need to make a face-grain plug of oak using a plug cutter that matches the size of the hole. Glue and pare it flush, stain and finish. Or create a fake keyhole like this: 

Looks to be close to a full inch hole, so not likely to find a plud cutter that big.
A hole saw with the center drill removed might work, but not freehand; drill press required

If you don't have a drill press to use a hole saw without the pilot bit, you can use the hole saw WITH the pilot bit to drill a hole in a thick piece of scrap, then remove the pilot bit, clamp the scrap to the piece that will provide the plug, and use the scrap as a guide for the pilotless hole saw.

Jason

A few suggestions on your quest to plug the hole.  First, determine exactly which type of wood was used for the drawer front.  Then look for a source of that type of wood.  Your best end result will come from finding a piece of wood where the grain closely matches the grain surrounding the hole.  A  lathe might be a better choice for making the plug.  Bear in mind that a 1" hole saw makes a plug that is about 7/8 inch in size, so you may need an 1-1/8" or even a 1-1/4 hole saw to get a blank slightly larger than the hole that you can trim for a precise fit.  If the hole edges are not clean and sharp, a fitted plug is still going to be pretty obvious unless you make a slightly larger hole with clean edges.

Added later 6 d 7 h 12 min 53 s:

What did you decide to do about the hole in your drawer?

Maybe this discussion is old and you have found a solution already. I saw your picture and I thought of the wood plugs that are used to cover the holes near the base of bolt down newel posts on staircases. You could drill out the hole in your drawer front to the correct size and then put the plug in with the grain oriented in the right direction.