The mug plotter is loosely derived from Evil Mad Scientist's Egg Bot - it uses the same control board, and a modified version of the software, so you can design and print your mug from inkscape.
To build the mug plotter, you're going to need the following as a minimum, regardless of how you approach it:
two stepper motors
5V Power supply
To build the Mark I mug plotter along broadly the same lines I have, you'll also need the following:
Something to cut it with (laser cutters are good)
250mm of threaded rod, and a nut to suit. I've used M5, but its not important.
Some 1/4" aluminium rod (but anything in that range will do).
Around 20mm metal tube big enough to slide over the rod - I didn't have anything to hand to drilled a 6.5mm hole through a bit of 1/2" aluminium rod.
Some connectors suitable to join your stepper motors to the threaded rod and 1/4" rod.
Lots of small (M3) screws and nuts.
There's quite a degree of flexibility in the choice of stepper motor. I went for Bipolar, 200 Steps/Rev, 28x32mm, 3.8V, 670mA because it was pretty much the first one I saw, but it's within the ballpark of what you need