To remove small, broken screws from wood, there's an ingenious gadget called a screw extractor.
No, not the always breaking, never working "easy out", but a miniature hole saw, which removes a small cylinder of wood with the screw inside it.
Here's a pricy commercial version, with very good pictures and a video showing the use.
http://www.stewmac.com/Luthier_Tools/Types_of_Tools/Routers_and_Bits/Bits/Guitar_Screw_Rescue_Kit.html
I would like to make my own – filing saw teeth ain't rocket science.
The main requirements are that the extractor-saw is big enough to go over the screw stub, but no bigger, and that the interior diameter of the plug-cutter matches the external diameter of the extractor saw. It may be possible to tweak the sizes a little by setting the teeth a little.
For larger screws, a material source recommended is a second hand stepped golf-club shaft, which gives a range of sizes!!
But I'm dealing with very small woodscrews on a guitar (#3 x 3/8"
.
I have seen people recommending roll pins as suitable – nice steel, available in many sizes.
http://www.drapertools.com/product/63943/120-Piece-Roll-Pin-Assortment
(Chinese sets off ebay are much cheaper…)
But my "dream" material would have thinner walls.
Can anyone put their lateral thinking cap on and suggest something?
BugBear