Hi Alec,
Dont forget the power taken by the arduino. I have just measured the current taken by my arduino uno. It is about 50 mA So 0.05 x 24 x 60 (Days) = 72 AH. With the ATmega328P removed from the board it still consumes about 30 mA. So even if the code was written to keep the cpu in sleep mode for most of the time quite a large battery would be required. This could be reduced by building the microcontroller onto a board with only the components required for the timer function.
Les.