How can I drill a hole in a small stick of wood?
This sounds like a long pencil with a wire instead of a lead.  There are a number of videos and pictures showing how pencils are made on the web.  What it boils down to is that first you make the holes (grooves actually, since you do it in halves), and then you make the sticks round.  Much easier than the other way around.



If you need to make a lot of them I think it will be safer and more efficient to do adapt the pencil manufacturing method to your job.