Wolfgang is correct about the power supply may be week.
The BPI-M3 actually requires about 21W to operate properly.
You can manage to get it to startup from the emmc after flashed with a 2.1A supply but you will have poor results.
I found that the minimum is about 12W or 2.26A at 5.3V
Previous Post Specifically about this
The reason you may not see a LED illuminate is that the BPI-M3 powerIC does a very fast check to see how much load the incoming power supply can provide without dropping voltage down too low. This test takes place on power up before any of the LEDs are turned on.
One last thought…
The OTG version of the board the one without the power in jack can only handle about 9W on the OTG so it is best to power the board from the GND and V+ on the pin headers.