In general DEAD, DRY cottonwood works quite well and (I think) grows all over the country (under the proper conditions).
I have made an ember with a cottonwood fire-board and spindle.
Try using a cotton cord for your bow. I think it slips less than some other cord. The bow, fire-board, spindle and socket are the easiest parts to construct. The tinder bundle may be the biggest challenge. Cottonwood bark can work. We have used shredded Juniper bark for our tinder bundles with great success. Sometimes I have gotten an ember but was unable to get a flame from the tinder bundle with other materials. I think you should work on the tinder bundle first and then work on the fire bow parts. Try putting a neckerchief on the ground and shredding your material over it. The really fine material that collects on your neckerchief will be key to catching the ember into a flame. Don't forget to put a piece of bark or your knife under your fire-board notch to catch the ember so you can pick it up and place it in the tinder bundle in the spot with the finest tinder. Then wrap the tinder around the ember and blow into the bundle. Make sure the tinder bundle is not too tight. It needs to be kind of loose so it can get a lot a air. Locking your wrist against your shin is key to keeping the socket from moving. Take nice long strokes with the bow. Keep going for another 10+ seconds after you start getting smoke. When you stop and the ember keeps smoking you are good. Once you have an ember there is no big hurry; the ember will keep burning for several seconds. You now have punk; like what is used to light fireworks.
So far my favorite parts are: Saguaro fire-board, Yucca spindle and a Juniper bark tinder bundle. I can get an ember quickly and easily blow it into a flame with these plants.
For store bought parts I have found that a ceder board makes a good fire-board with a pine dowel. Take toilet paper and shred it to make a tinder bundle. Almost anything will work for the bow and socket if they are the right shape.
Be sure to put some Chap-Stick or something in the socket to reduce friction.
Making fire by friction is one of those things that seems like a waste of time; but is a great emotional experience once you make it happen.
Good luck!