The first plan was to use a laboratory heat/stir plate to both heat the water and stir the coffee. Simple, elegant, one moving part. To ebay!
Unfortunately, the hot plate took over 20 minutes to boil the water and required taking the French press beaker out of its holder.
Plan B was to use a percolating coffeemaker mechanism to deliver near-boiling water to the grounds, but testing showed that the temperature was significantly lower than the optimum 195-205°.
The right answer turned out to be a Sunbeam "Hot Shot" water heater. This $15 device, with its 1450W heater, boils two cups of water in 30 seconds flat -- faster than a stove, microwave, or electric kettle.
In stock form, a manual lever actuates a poppet valve. I hacked a solenoid to press the lever as a proof of concept.
The solenoid was strong enough to dispense the water -- barely. I researched solenoids and learned that they produce the greatest force at the end of their travel, so I used SolidWorks to design a linkage which would use the full travel of the solenoid to open the valve.
For aesthetic reasons, I knew I didn't want to keep the Hot Shot's plastic housing. I also needed a place to mount the solenoid and a pivot for the new lever. I'd used machined parts for many previous projects, but I hadn't done much with sheet metal so I wanted to use it here.
The bracket is 6061-0 which I chose mostly because I didn't have the correct gauge of steel handy. The untempered aluminum doesn't crack when bent. Although it was totally unnecessary, I used FEA to optimize the bracket for weight. (Also note the gratuitous use of the 45° chamfer mill to break the edges.)
The lever fork is also sheet metal, made from two pieces of galvanized steel milled on the CNC mill, bent into shape, and then soldered together with silver-bearing solder. The seam between the pieces is almost undetectable, though there are more pliers marks than I'd prefer...
With the Hot Shot installed, the lever is self-retaining. The improved pivot position allows the solenoid to easily open the valve fully.
Given the high voltages and temperatures at the heater, and aesthetic requirements, I'm currently working on wrapping the heater in a thin stainless shroud and adding a plastic water tank, as shown in this SolidWorks screenshot: