Depends on whether you want a "light" or "dark" look ....but on the light side I'd go with Bird's Eye Maple ....in between I'd go for a natural Cherry look...and on the dark side, I like Black Walnut. I like the "natural look" of all 3 woods..and personally I'd use a clear lacquer finish on them ...like Deft.
I saw one of the notes above mentioned spalted wood...but spalting is the first stage prior to rot ...and it gives you some amazing colors from the staining.../ but a lot of spalted wood tends to be "punky" and very difficult to stabilize....not impossible / but tricky. If I had some "spalted maple or cherry" I'd try it ....but personally spalted woods tend to work better for ornamental pieces .... vases, boxes, etc...vs something like a handgun stock.
Air Drying your own wood blanks can be done...but its best done in some kind of a kiln.../ and air drying can take a long time ...and the guys that do a lot of it ...will tell you a board one inch thick will take at least a year. If you really want to salvage wood ...cut it into some rough blanks...2 or 3 times the size you think you'll need for the finished product....coat the end grain to help prevent cracking and hope for the best ( and expect to lose 30% - 50% of your blanks to cracking ). The more "burl" ...or the more crazy grain look they have ...the more difficult they are to dry without cracking...but the more outrageous they turn out too ...
You could use thin pieces of wood ...that you might cut on a bandsaw ...and then laminate them together to form a blank. When you cut thin pieces ...it tends to relieve some of the stress in the wood...that will cause cracking ..so its a way to defeat the cracking issues. You could also use 4 or 5 different species ....like maple and black walnut ...and laminate them together for an intersting look. It would be unique.../ a lot of guys will do it on things like fancy boxes or for items like peppermills...
I'd suggest you experiment ...first with some commercial woods ...buy some blanks of maple or cherry ...and see how it goes / shaping, sanding, etc....and consider how you want to checker them ..or not .. / checkering is the most difficult step...deep enough to give it character..but not deep enough so they crack...and it can be done using a router, handsaw, bandsaw, etc....depends on your skill - and what you feel like trying.
Kind of like the first 10 peppermills I turned...it took a while to get my techniques down...and my designs fine tuned. Its not that they were bad ...but there is a learning curve on all of this stuff ....but that's why its fun too. Hope you give it a try...