re: Grandpas rocking chair. What would you do?

Is the wood varnished or shellacked, or simply sealed? Or is it polyurethane? In any event, you can clean the wood parts with Murphy’s oil soap and water, dry thoroughly, then use either a good quality paste wax (like Butchers). If the finish is completely gone in places, you could use a danish oil finish (a penetrating linseed oil), buff by hand with steel wool and soft cloth, then wax over it. ETA I like Tried and True brand danish oil finish.