You can purchase masonry anchors that will support a lot of weight (it will say on the package how much).
Command hooks might work. You can drill into the stone with masonry bit or use masonry nails but that will put holes in the walls.
If you don't want to damage the wall look into command hooks. If you don't care drill and insert plastic anchors and then the appropriate screw.
3M Command hooks
I would use a masonry bit and drill a small hole for an anchor. If you change the location of the anchor, they holes are simple to fill.