Powering an Arduino Uno using 2 (or as few as possible) AAA batteries
I've got a page about power saving.

My first suggestion would be to not use a Uno, which is quite heavy (because of its size) but a smaller Arduino like a Mini. Second, I think you'll find that LiPo batteries are quite light, and come in various sizes (and thus, weights). A single cell will output around 3.7V which is fine for driving an Atmega328P. Possibly you could drop the clock to 8 MHz because of the voltage.

If you don't need it to be "awake" all the time consider using the techniques described in my link above to make it sleep for (say) 4 seconds at a time and then take a measurement.