I'd be interested to see other replies to this - I also have an Arduino breakout and am planning to build a project with solar / Li ion battery.
At the moment I'm working on the assumption that it needs 7V – 15V on the DC input with 500mA+ (per Sparkfun's page) for the board itself ie. not including sensors etc.
The Arduino Expansion Board can be powered through the DC barrel jack or through the USB client port. The first option requires an input voltage from 7V to 15V, and the second option requires a 5V input voltage. The difference between these two ways to power the board is when the Host USB functionality is used. It's necessary to power the board through the DC barrel jack when the Edison is working as Host USB.
Regarding the 3.7 LiPo battery, I haven't tested it but it seems to be possible to power the Arduino Expansion Board through the USB client port if the voltage is converted to 5V first. I think you will find useful a boost converter. You can learn more about boot converters here: Boost converter
Thanks DiegoV_Intel -- I already have a 5v ready solar power supply that I built based on this writeup from Adafruit. I've been using it to separately power a set of servos by routing the power through a booster, also from Adafruit.
Below is what the setup looks like, I just need to diy a male usb header to give it a shot. Speaking of which, the USB client port is the full-sized USB on the board, correct?
It works! I didn't use the full solar setup since I'm waiting for some jst wiring to show up in the mail, but I was able to test it with the battery by itself for now.
Photo of the setup (Adafruit Powerboost + lipoly battery + short usb cable)
I've read around that the edison's power usage can hit 6-700mA when communicating over wifi or using a microSD card. I didn't have a problem with this powerbooster, & working via ssh at 500mA, but to be safe I'll be using it with a 1000mA booster that's currently in the solar setup from above.