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
Step 1: Assembling the mug mounts
Cut the 1/4" rod to length - you'll need:
Start by assembling the mug mounts - connect the two concave frames together, and then the convex two. Place a circle frame on the end of each to keep it square, and then attach to a piece of your 1/4" rod with a bit of glue - 25mm on the convex one and 80mm on the concave one. You need to file down the last 10mm or so of the 25mm rod to provide a surface for the connector to grip to. Cover the sloped ends of the two mounts with some draft excluder tape, and slide the spring onto the end of the concave mount.