Materials:
Carpet roll - call some local carpet stores; you only need one, 'cause one will make like 20+ drums
Saw or razor blade
Drill and drill bits
Sturdy string (I used kite string)
Durable plastic "fabric" or leather
Needle
Paint (optional, but way more fun)
1/4"-1/2" thick rope
Take a saw or razor blade and cut cylinder in 10" segments. Around the top edge, drill small holes just big enough for a needle with your string to get through about 2" from the top. Just above the middle on opposing sides drill a bigger hole that the rope can just barely fit through, if you're doing the neck string.
Paint it if you want; put on a non-toxic sealer if you want to make sure the kids won't lick off any paint if you're worried about that sort of thing. Be careful of the holes or just redrill after the paint dries. If you just try to drill after painting, you'll probably crack the paint. I know because I tried.
Cut out a circular piece of plastic/leather that's about 3-4" wider than the tip of the cylinder. You'll sew the plastic/leather on through the holes you drilled. Make sure it's tight. Tie the ends of the string together. Put the rope on so the ends are on the inside. Make a knot so it can't get back through. Sand yourself up some dowels and you have drum sticks.
Have fun drumming!