Deidre, I believe that I would drill out the hole to a larger size and make a plug out of oak with a similar grain to fill the new hole. Without seeing the piece, I'd hate to say more but this should not be a big job to an accomplished woodworker. Restoring the finish to match the current finish will be the biggest problem, probably. Many of these were painted originally. You might like to check out the picture of the ones made in Grand Rapids, MI in the book on the Grand Rapids furniture manufacturing history.