Some notes on component choices
MIcrocontroller: I chose the Attiny85 since it is cheap, small and can be programmed with the Arduino IDE.
OLED: I would have chosen a smaller OLED display than the 0.96" on a breakout, but it is just larger than the CR2032 battery holder used on the back, making a neat package.
Load switching mosfet: The Vishay SI2318CDS mosfet is used because it has a low RDS(on) (drain-source on resistance), to prevent excessive voltage reading error when the load is switched on. The gate voltage is also low enough to be switched by the coin cell battery (2.5V) and is in the very common SOT-23 footprint, allowing me to subsitute another MOSFET if required for any reason.