The electronics are the easiest part of this endeavor, thanks to Analog Devices, Inc. The AD7745 IC is a 24-bit fully differential capacitance to digital converter. It is highly integrated, so very few external components are required. Here is the entire circuit schematic.
My original design used a LD2981 3.3V LDO instead of the XC6206. I decided to save a few pennies with the XC6206 since the current loading of the AD7745 is less than 1mA. Total cost of the electronics is about $12.25 -- the majority of which is the AD7745 ($11.28). The circuit board connects to the pin header of the Raspberry Pi, but hangs outside the Raspberry Pi to avoid potential interference problems.