Yes, the Edison Breakout Board can be powered using a Lithium battery. According to the Breakout Board Hardware Guide, you should attach the lithium battery to J2. Also, if you power the board via J21, J22 (power jack which must be installed first) or J3 (USB) the breakout board will recharge the battery. The battery should provide 3.7V and at least 300mAH capacity.
You should check the Hardware Guide for some more information on this.