In terms of efficiency, this is about as good as it will get. The most efficient method of transforming the voltage from 12V to 5V is to use a buck converter as opposed to a linear regulator. The UBEC is a buck converter. For power levels in this range, an efficiency of over 90% is considered good.
If you were willing to search around for a custom buck controller IC and do some PCB design, you could potentially eke out up to 96% by selecting an IC that includes a technology called synchronous rectification. An example of such an IC is https://www.fairchildsemi.com/ds/FA/FAN2001.pdf (however note that the input voltage on this IC only goes up to 5.5V)
With synchronous rectification, you can still achieve a good efficiency at low current levels (see page 5 in the pdf). For 100mA you can still get well over 90% efficiency.
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
Best Answer
If the sensor is supplied from the same voltage source as the microcontroller, generally there is no problem. It's not a matter of the board being "ready" so much as never applying voltage to an input that exceeds Vdd of the microcontroller. The actual specification is:
So if the IO supply voltage is 0V the input voltage should not be more than 0.3V or less than -0.5V.
If such a voltage is applied without limiting the current, it's possible to damage the microcontroller.
Alternatives include adding some series resistance to limit the current or buffering the input with "something" tolerant of voltage applied when power is off. There are digital buffers that are designed for this purpose (they also translate voltage levels), and for analog circuits you can use an op-amp buffer with some input resistance to protect the op-amp.