I sometimes use a piece of wood, for a box lid, from the same stock blank the gun came from, but oftentimes I don't. I just go to the scrap bin and pick out something I think will look decent when finished.
With regard to a wood box, keep in mind it's best to plan for it in shaping the stock. That is, a flat is established and wood shaped to it rather than the stock shaped and then a flat created. It can be done after the stock is shaped, but doesn't work out quite as well in my view. Not sure how far along you are.
Good luck,
Jim