An all-time favorite project, a weather station. But this time the plans include an 100% autonomous outdoor unit.
The above require a LiPo battery and a solar panel and an Arduino Fio.
Fio consists of an ATmega328P (the same micro as the UNO), a connection for a LiPo battery, a charge circuit based on the MAX1555 (with an USB connector) and a socket for an XBee radio module. So all you have to do is wire things up and work on the code to reduce the power consumption to its minimum.