Like everything else around this place specifics help. Here are some generalities. English walnut (AKA all kinds of names) is generally considered the best stocking wood. It takes and holds checkering well. It is strong for it's weight. It can have beautiful figure. The perfect piece will have similar figure on both sides of the buttstock and forend while having straight grain through the hand.
Next you have to figure out how you want to finish out the stock. Do you want a cheek piece? If you want a cheek piece, do you want a shadow line added? Do you want a straight comb or a Monte Carlo? How do you want the butt treated? It can of course be a recoil pad but also a skeleton butt plate with checkering, or a checkered steel plate, or straight checkering only, etc. How about a forend tip? Ebony is the classic option. Checkering the next big decision and it has infinite options. How many lines per inch, fluer-de-lis, borders or no borders, ribbons through the checkering pattern, etc. The more complex the demands become the fewer people there are who can execute it properly.
If you have any interest in these kinds of extravagances you should be measured for fit. There are a fair number of guys competent to fit shotgun stocks but not so with rifle stocks.
There is another way to do this. You become your own general contractor. This company is well known in the shotgun world. They are very knowledgeable and fairly priced. Choose a blank with good figure in the butt and most importantly good layout in the hand.
http://www.gunstockblanks.com/faq/q...and-payment/how-much-do-your-blanks-cost.html They can also suggest stockmakers. This women is/was a well known checkerer.
https://www.checkering.com/ As far as who to do the work it's always the same. If you choose the most noted stockmaker you will have a long wait and a big bill. My interest in this area of gunmaking is from quite some time ago but there are always stockers who do nice work at a reasonable, comparatively, cost.