Current shunts are just high wattage low value precision resistors. On larger units the current through the shunt could be as much as 1000A and would result in a 50 mV drop - 50 microohms. Not knowing the rated current of this motor, it's hard to say which range is best but a 10 millohm shunt would drop 50 mV at 5A. If the motor full load current is 20A, the shunt should probably be smaller - like 0.05V / 20A or 2.5 milliohms.
There's nothing magic about the 50 mV spec except that it is commonly used in industrial systems. The meter manufacturers just produce one meter range (0-50 mV) and various scales for the application. 1000A, 100A, 1A - doesn't matter. It's the same meter with a different scale. Only the shunt resistor is different.
Using a DMM, there is truly no magic in 50 mV. It doesn't matter what the drop is as long as it is insignificant to the circuit. But the voltage drop must be high enough that it can be measured and the comment about the Brymen meter is that it will measure to 0.01 mV. Some meters have better resolution, some have worse. A bench meter might measure to 0.001 mV and my HP will measure to 0.0001 mV or 0.1 uV.
When using high side current sense amplifiers, the maximum mV range will be in the datasheet and then it's simply a matter of picking the right resistance. The range should probably include stalled rotor such that the amplifier isn't overloaded during motor startup.
The Brymen has MIN-MAX-AVG recording and AVG might be just the thing to smooth out the readings during an experiment. Many meters have this feature, it's not uncommon although it is often overlooked. MAX may also be useful.
The BM237 is just a 6000 count meter. It has decent accuracy, great safety ratings, and Dave's required uA scale. Other than that, it's nothing particularly special. I bought a couple of them simply for the branding and to support Dave's efforts.
It is currently unavailable at Amazon:
https://www.amazon.com/EEVblog-BM235-Brymen-Multimeter/dp/B01JZ1ADCOShorter answer: You need a way to measure the current to a satisfactory resolution such that you can see the difference in current between having the gearbox installed and not having it. The measurements need to make sense.