Wood Veneer - make it something special, this is a great chance to use something really exotic. My olive ash piece had a really great pattern. It needs to be just a straight piece of veneer, none of that paper backed stuff they sell at home improvement stores. A 2x12" piece should get the job done.
Wood Base Material - I had 1/8" thick plywood laser cut to build the backing of the bracelet out of. If you do any woodworking you almost certainly have a better selection of tools to work with so you can probably make this with a jigsaw, bandsaw, router or some other clever way. The upside to lasering is that I was able to make parts for another project from the pieces cut out of the bracelet and I was able to be super precise.
LEDs - I used the impossibly tiny surface mount style so that I could keep the channel as narrow as possible. I didn't want a big, bulky bracelet. If you're willing to go with a bigger bracelet you can use much larger/friendlier LEDs. Mouser #630-ASMT-TWBM-NT902
Resistors - One per LED, use a handy calculator
to get the right kind. Mouser #299-150-RC
Battery Holder - This needs to hold 2 coin cell batteries. Mouser #122-2820-GR
Coin Cell Batteries - I used CR2016 batteries because they're only 1.6mm thick and took up a small amount of space. Mouser #658-CR2016
Resin - A two part epoxy resin is ideal for this. You can get away with the table top stuff if your channel is thin, but a casting resin is better.
Resin Dye - I used a small amount of white resin dye to help diffuse the light.
Wire, Solder - Use this to connect the LEDs and resistors to the battery case. Lead free is best, but I prefer solder with lead. There are really only two tiny areas that they lead might be accessible, and you'd probably die from ingesting all that resin, wood and varnish before the lead got you. I stripped the insulation off of some stranded copper, solid would have been a lot easier.
Wood Glue - To glue the wood together. Keep it closed and keep the nozzle clean so you don't have to fight dried bits.
Barge Cement - To glue the leather to the wood.
Rubber Cement - To temporarily tack parts together.
Super Glue - To quickly stick the circuit to the bracelet. It only needs to hold until the resin gets in there.
Clasp or Closure - This will close the leather over the battery pack. I used two small snaps.
Sturdy Needle and Thread - To sew on the snaps onto the leather.
Wood Finish - Some sort of varnish to go over the whole thing with. I like water based varnish. I'm sure you have a favorite.
Leather - This will line the bracelet. I used some factory scrap from an outlet. Consider what color will look good and what will feel nice on your skin. Thinner is better.
Soldering Iron/Solder Sucker/Sponge - To solder the electronic goodness together. WIth a little luck/skill you won't need the solder sucker.
All the Small Clamps In the World - You're going to be forming wood veneer. More clamps equals more likely success. The cheap brightly colored plastic style will get the job done, no need to pay up if you don't already have some. I've picked up tubes of a dozen or more for a couple bucks before and they're endlessly handy.
Resin Mixing Gear - Paper cups and popsicle sticks for easy mixing and pouring.
Scissors and/or a Utility Knife - To cut the veneer.
Container for Soaking Wood - This isn't for food anymore, and it can take some heat.
A Way to Heat Water - I have a handy electric skillet for this, pick your favorite. Hot tap water was enough to soften my veneer, but you might need more heat.
Gloves, Dust Mask, Safety Glasses - Don't play too cool for this stuff. Resin can have permanent affects on your skin. You don't want a piece of flying shrapnel to be the last thing you ever see. Dust can have lasting consequences in your lungs. Don't say I didn't warn you.
A Way To Drill Two Small Holes - I used my Dremel with a 1/16" drill bit.
A Way To Cut Wood - I had my parts laser cut because I lack the equipment to do it at home, but if you can do it with your equipment then, by all means, do.
Sand Paper - A nice range of papers is best. The finer you go the nicer your bracelet will be.
Masking Tape - Useful for taping over places that resin might seep through.
Standard Household Items - Paper towels, typing paper, other ordinary stuff you probably have.