BeagleBone Black - power supply sensitivity
To try to get closure on this, and for my own sanity, I read Section 5.4 "Power Management" of the the BeagleBone Black System Reference Manual (BBB_SRM), Rev-C.1.  

It says BBB uses two devices to supply the board with power:
TPS65217C has a Vin (Max) 5.8V
LDO TLV70233 has a Vin (Max) 5.5V
These are operating voltages.

The BBB_SRM says the "external LDOTLV70233 provides the 3.3V rail for the rest of the board." 

However, searching the BeagleBone Black Schematic (BBB_SCH), Rev C, does not find the TLV70233.  

There is a TL5209 on the "BeagleBone Black Power Management" schematic. It is connected to an output of the TPS65217C. It is not connected to the input power socket, and hence isn't directly constraining the maximum DC voltage input. It provides 'VDD_3V3B', so I believe it is the LDO referred to in section 5.4 of the BBB_SRM.

The text of section 5.4 is the same as the Rev-A6 BBB_SRM, so I suspect the text of the BBB_SRM is out of synch with the BBB_SCH Rev-C.

Summary @Farham is correct.
The only active part connected to the DC socket is the TPS65217C, and it has a maximum operating voltage of 5.8V. So 5.24V should be fine.

I assume the BeagleBone Black engineers specify The DC supply should be well regulated and 5V +/-.25V to provide some headroom; the BBB power management subsystem will survive voltages more than 10% over nominal.

Edit: I have raised an issue about the inconsistency of LDO part number of section 5.4 of the BBB_SRM and the "BeagleBone Black Power Management" schematic BBB_SCH at BeagleBone Black github