The Rayovac alkaline batteries I've been using have enough life down to 0.8 volts (each times 2 batteries for 1.6V) to keep an ATtiny85 running reliably. In other words, I believe 1.8V is a good threshold for AA alkaline batteries.
The ATmega328P can accurately measure it's own supply voltage. You could set the BOD at 1.8V (or possibly disable it) and add a bit of code to "shutdown" (deep sleep) at a voltage of your choosing.
If the processor will spend anytime polling 1 MHz (or even 128 KHz) may be a better choice.
Probably not much if any, but are you thinking more from a power conservation standpoint, or immunity to the Vcc fluctuation?
Power conservation.