The problem is that the noise from the microphone will be much less than the noise from the pre-amp that it is connected to.
The usual way to measure noise is as “rms A-weighted”
There is a plug-in available here that can measure this: "Wave Stats" plug-in
Installation instructions are here: Missing features - Audacity Support
You can measure the noise from the pre-amp by replacing the microphone with a 600 Ohm resistor (the value is not critical - a 1k resistor will do). Make a recording with the recording level turned up and the recording will contain noise from the pre-amp. It is essential that the recording level is exactly the same for each test.
To make the measurement, select a short section of the recording and use the above plug-in.
Avoid the start and end of the recording as there may be clicks at the start/end.
To make measurements with a microphone attached you will need an extremely quiet room to conduct the experiment or the results will be meaningless.
Note that condenser and electret microphones have active electronics built in, so it is not possible to measure the noise level of the mic capsule alone.