I beleive the stick used the analogue port for the X,Y and Z axis, and was set up in absolute mode (position on screen proportional to displacement), rather than the relative mode of game joysticks (displacement determines speed of movement). The approximate 10bit accuracy of the Beeb's ADC would have been fine for hitting each pixel on a 640x256 screen, but would require quite a bit more accuracy for the higher res screens used today. I don't think it was self centering making driving it in relative mode difficult, and you'd only end up with a glorified laptop nipple which doesn't suit everyone.
If the mouse is becoming a problem try a graphics tablet, as holding a pen can be more relaxing for some types of operation such as clicking and dragging, and of course is particular good for drawing graphics. For other things such as drag and drop with buttons held at the same time its more difficult, and the mouse still wins out for general desktop manipulation.