Expected value of a quantized noisy signal
This will should work fine.
For each realization $k$, we can write
$$y_k[n] = x[n] + w_k[n] + q_k[n] = x[n] + v_k[n]$$
So basically define the &quot;effective&quot; noise $v[n]$ as the sum of the analog noise and the quantization noise $q[n]$. Averaging will converge towards the desired signal $x[n]$ as long as two conditions are met:

$v[n]$ is uncorrelated with $x[n]$
$v_k[n]$ is uncorrelated with $v_m[n]$ for $k \neq m$

So as long as the quantization doesn't create any correlation with the desired signal or across realizations, your good to go.
In general this will always be the case if the analog noise is significantly larger than the smallest quantization step and also uncorrelated across realizations. This will even work with a 1 bit quantizer, it just going take a really long time to get any decent signal to noise ratio.
EDIT
The first condition will NOT be met if the analog noise is smaller than the quantization noise, i.e. $  \sum w_k^2[n] &lt; \sum q_k^2[n] $
In this case the quantization noise dominates and is indeed correlated to the signal. You can just dither it: add more noise to decorrelate. The choice of the dither is primarily determined by the quantizer and not by the signal itself. If you know roughly the spectrum of your signal you can throw in a noise shaper as well to help things go faster.
It's a bit counterintuitive: by adding noise, the final effective quantization noise gets smaller.