You should probably feed the 9.6V into a voltage regulator and get out a stable 5V output and build your circuits for 5V. This will make sure that the battery voltage can drop to about 7V before you see the effect on the display. Also remember that the Arduino does not take 9.6V. There are lots of circuits for chips like the 7805 which make building this power supply really easy.
800mAh does mean 800ma for 1 hour.