How can I power Beagle Bone Black with 3.7V LiPo?
The BBB has a specific, unpopulated, header just for a single cell(3.7V) LiPo, charger and all!  It sits just behind the 5VDC barrel jack, 4 pins labeled TP5-8.  Page 41 of the A5A manual shows the pinout. Keep in mind that powering from a 3.7V Lipo via the battery pins will prevent the USB Host port from providing 5V power.
This guy has a small tutorial on how to wire it up:
http://www.element14.com/community/community/knode/single-board_computers/next-gen_beaglebone/blog/2013/08/10/bbb--rechargeable-on-board-battery-system

The BBB has a built-in power management IC (PMIC) based on the TI TPS65217C device. This device contains multiple switch-mode regulators and LDO regulators to provide all voltage levels needed for the entire board.
The IC also contains built-in battery charging capability.
Apart from the USB requirement of 5V, there is no need to run the BBB from 5V; it can happily run from a lower supply. This means that a single 3.7V cell could be used to power the entire board. No need to step-up to 5V or to run dual cells and step-down to 5V, both of which could have been inefficient.
The BBB has four holes that are suitable for connecting up the battery. They are detailed in the BBB system reference manual (SRM)


Now I personally was able to power it up omitting the 10k resistor mentioned in lieu of a thermistor, where Batt+ goes to both TP5 &amp; TP6, Batt- to TP8.  The thermistor (or 10K resistor bypass) would have been between TP8 and TP7