The design of this requires heat to transfer from the side of the mug to a mounted heat sink. The original vision of this design is to have the heat sink mounted on a surface that is 90 degrees rotated from the surface that will conduct heat from the side of the cup. For the best power output I'd require maximum heat transfer from the mug to the thermo electric generator and heat sink.
This is a good comparison of readily available metals and their heat transferring ability (in the easy to understand and fun BTU per hour foot degree farenheit!). Copper is the ideal choice, but pretty hard to work with for the average person. Luckily I have a very talented friend who was able to help me bring this vision to life. With some dimensions and a terrible drawing he was able to make it happen.
I ran a quick test with a hot mug of water, holding the copper, the thermo electric generator, and the heat sink together. I was able to generate an impressive 150mV, which I believe may be enough to step up to a usable voltage for a fan. Being able to run a fan will aid the heat sink in dissipation, yielding even more power from the generator.