Hi David,
Not sure what has been said so far, but I do find myself buying a new dremel almost every year or so (old ones still work fine, just being overly cautious with trying to avoid potential cross-contamination, etc.) and I find that buying the Dremel at ~£60 (with cord) to £100 (cordless) is easier in terms of finding them in the shops, but a few years ago I did buy a nice 100 piece kit with Draper drill included for less than £30 total from a local random store which worked just as well.
As for the drill heads themselves, yes the dremel ones can be expensive, but in shops like Clas Ohlson (not sure if I should be advertising on zooarch list :) ...) you can buy diamond point grinding sets that give you 20 diamond-tipped drill heads (3 mm shank) for ~£14 (Value Tools - Diamond Point Grinding Set 20P), much much cheaper - the only catch being that they come as a variety of shapes, some of which you might not want to use, although most are fine. Don't use the circular blades myself, so if you are cutting pieces out of the bone rather than drilling for powder not sure if there is a cheaper set of these available, but presumably there is.
At that price (<£1 each) you could even justify a fresh drill head for every sample if need be depending on sample number and nature of the project.