Thanks for the reply!
I did get the datasheet for the 8211, but I'm still having trouble understanding it. There are multiple examples, and all the circuit design terms confuse me 
I also found some Maxim "fuel gauge" battery monitors, which give estimated remaining charge in % and mAh. They looked perfect for what I want, but they're only available in surface-mount packages, not DIPs like the MAX8211.
The voltage divider and analog-in approach seems more desirable, because it's an analog reading, not just a 1 or 0. I would probably have the Arduino-powered device send it's battery voltage via serial to a computer, which would log the voltages until the thing dies (one-time calibration). Then, that value would help the Arduino know when it is going to run out of power.
kg4wsv, does that sound good? I'll look into the voltage-divider thing. Thanks!