There are different ways to power the mini-breakout board. I explain each one below. Check the following picture taken from the mini-breakout board schematic:
J21 header: This is the way you are currently using. The breakout board can be powered through this header with a power source from 7V to 15V. You can easily power the breakout board if you add a female 2pin header (check picture below) to your power supply cables so it won't be fragile as you mentioned.
VIN input: The VIN input is located in the pin J17-4 of the breakout board. It's hard to power the module through this pin unless you have designed a custom board for it. You can power the module with a power source of 7V to 15V.
J22 DC barrel jack: This third way is optional and it will require adding a barrel jack to the breakout board. The board can be powered with a power source of 7V to 15V using this way.
USB power: This might be the easiest way to power the mini-breakout board. It uses the USB OTG port to power the module so you only have to connect the board to your PC with an USB cable. This power input accepts only 5V.
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The Intel® Edison Breakout Board Hardware Guide gives a Digikey part number for the J22 DC barrel jack. It's for a 2.5mm post version though.
I recommend switching it to Digikey CP-002AHPJCT-ND which is the 2.1/5.5mm version of barrel jack with the exact same footprint. Using the 2.1mm version makes the jack compatible with tons of barrel jack accessories and power supplies from Sparkfun and Adafruit. They don't really offer anything for 2.5mm. 2.1mm is also the size of Gallileo power supplies which the user might already have.