My "old injun friend" specializes more in beadwork and other "arty" stuff, but has made a couple of traditional instruments, including drums. His method for leather covered beaters (from his description, not from watching it done) was just to wrap string around the end of a dowel of appropriate size, stretch a wet piece of leather over the end and tie off with string.

My impresion is that some "store bought" beaters may use more or less solid felt padding in place of the string padding.

Most readily available leather will have a "finished" side and a "fuzzy" side. For native drums, usually the beater would have fuzzy side out. Once it's dried you can, if desired, "scuff" the leather for a little softer surface, since it may be somewhat hardened by the soaking and drying; and of course you probably will want to trim off the wrinkles and excess string. A fairly fine cheese (or garlic) grater may work for any scuffing you'd want.

Variation: Use rawhide boot laces instead of string, both for the padding you wrap on the stick and to tie the leather cover on. After soaking and drying, the rawhide padding may be pretty "hard."

Variation: Cut a few grooves where the string/rawhide will wrap on the dowel for better retention.

Variation: Use the string to assemble, but once the head is dried trim off the string used to hold the leather and replace with a fancy (colored or even beaded) string, or even with (colored?) wire.

The string/rawhide padding under the leather should be something reasonably "compressible" if you want a softer head, so if you use string a cotton/linen twine may be better than nylon or poly.

You can "smooth" the butt end by pushing it in the dirt and "twirling" it between the palms as if you were starting a campfire. If you want a better "feel" to the stick you could start with a snare drum stick with diameter and taper and such that suits you. Trap players I've known usually buy in pairs, and break sticks fairly often (they've been mostly young rockers) so "half-a-pair-freeby" sticks shouldn't be too hard to find.

The process is pretty simple, but the leather cover needs to soak for a couple of days to be really workable, and it may take a week to dry fully. (If not done with some care, there could be mildew.) Tying off and trimming may take some practice before really "neat" results can be expected, and if you try out different decorative effects you might end up being called an "artist."

If you're just into playing, buying your first beater(s) might be the best route, but making your own could be fun to play around with, if you're inclined to being a bit "crafty."

John