I wouldn't touch it. Keep it oiled to prevent rust, and avoid getting gun oil on the stock, it will break down the wood fibers and one day you'll shoot it and the whole action and barrel will break loose from the stock. If you decide to refinish it, take it to a gunsmith or do alot of research first. While spray paint (KRYLON) might give a good finish, its paint. It will never look like wood again.
There are some good videos on youtube about stock refinishing. I wouldn't use wood filler on the craters. As long as the grain was not cut on the wood, you can steam the dents and dings out using a wet rag and iron. This will also help remove some of the old stain. Use steam or something like crud cutter to remove the old finish. Be patient. It take time. After the stock is stripped and dries progressively sand it with soft abrasive pads like scotchbrite untill you get a nice smooth finish with a very fine grit pad. Then pick your choice of stain or a nice oil finish. If you aren't confident you can do the job, don't do it. I think the sentiment of your grandfathers gun with all of the marks from where he used it will mean alot to you, Way more than a ruined stock.