use lm386 - the simplest
The LM386 is a power amp and does not have good noise performance.
OP
Given you're using a 5v reference. 1 lsb = about 5mV
The noise you see is about 5 or 6 lsb p-p ie < 2 lsb rms or 10mV. What was the sampling rate?
The gain of the opamp varies between 11 and very large depending on the pot setting - where was it set?
The impedance of an electret is probably 1 to a few k ohms.
The universe gives you -174dBm / Hz at 300K ambient temperature.
The circuit you show has some crude but unpredictable HF rolloff - any idea what the -3dB rolloff point is? - it could well be worth making this explicit. I'd add a 180 ohm between the top of R6 and C3 to give a 20kHz-ish rolloff. Given the 10k max samples/second of AT328 based arduinos it could be worth making this much lower - so try 820 ohms.
And add 100pF across R4.
Apart from anything else, many opamps don't like directly driving large capacitive loads - this makes it's life a little easier.
Those in the know will see I'm trying to get at a noise figure. For this I need to know the actual gain of the opamp.. ie the pot setting.
Can you help, OP?
Of course the microphone will itself have a noise output. Do you have specs for this?
Or try re-measuring with the mic replaced by a 1k resistor.
Allan