Using the Allwinner A13 is a really good way to go - an existing SoC with a good track record and, most importantly, frugal energy consumption suitable for use on battery power.
Their nice vid is here:
The World's First $9 Computer: Meet Chip by Next Thing Co.
Click "PLAY" on the banana.
By comparison Raspberry Pi is very much a high-consumption bodge-up, it sucks so much power that a single usb will not support it, and to use it on portable power you would need to tow a car battery on a trolley.