Heck, I'm an official woodworker and I don't think I have the patience to make a stock from a chunk of wood. But...if I was gonna do that, some wood choices are:
Walnut (you can buy that from specialty hardwoods stores or off the internet)
Maple (mostly real light colored, but can be stained as desired. Can be had in some fabulous grain patterns)
Pecan/Hickory (tough and pretty)
Mesquite (pricey and you'd have to be really careful to get a good piece of wood with no splits in it. Mesquite is bad about that).
Osage Orange (beautiful, but splits bad when it dries and really tough to work with)
Elm (very tough wood and would work for gunstocks)
Cherry (gorgeous and I think it's tough enough for a stock)
Ash (ought to be Ok)
Redwood (I think no)
Then you have a zillion choices on imported woods, some of which are pretty enough to put tears in your eyes (Rosewood, Bocote, and Australian Blackwood). I made a rolling pin for a friend and did it in Blackwood. Prettiest thing I ever made, even if it was just a rolling pin. I guess that work of art is still in use rolling tortillas.
As for how I'd go about it, first I'd check the internet for ideas, but if I had to start today I'd take the old gunstock and measure all the dimensions. I'd take the new piece of wood and cut it to within 1/8 of an inch into a rectangular block of wood (maybe 6 inches tall, 3 inches thick or less, and whatever length is needed. Then sketch out the side image of the stock on the wood block and cut to within 1/8 or 1/4 inch in all directions. Now you have something that looks very roughly like the shape of a gunstock, and now the real detail work begins. The outside has to be shaped to what you want and the interior needs to be cut away as required. You need very sharp chisels and a steady hand for the inside work. A grinder and random orbit sander and belt sander can be used for roughing the outside. And about this point in the project, I'd probably call Boyd's, Stocky's, or Hogue. You gotta really want to do this.