Will noise floor dominate the ADC resolution?
The short answer is that - in data acquisition - you should typically use the smallest full range that accommodates your signal + noise. The RMS noise should be always much bigger than the Least Significant Bit (LSB) size.
What you discern as &quot;1 mV noise&quot; can contain a great deal of information that could be lost to discretization effects if you used a larger LSB size. DKNguyen's comment suggested this. Given the signal is stable over time and the noise is white, you can use averaging or Delta-Sigma modulation to obtain a reading with virtually infinite precision as long as the useful signal is sufficiently dithered by noise.
The more noise you see, the better for accuracy. If your RMS noise gets even close to the LSB size, you start to see aliasing and this will alter your reading and look e.g. like a fake offset. The limit where spurs become a serious concern is a noise of 1 LSB. If noise is lower than that, your signal is no longer properly dithered by its own noise. Even with slightly higher noise, this effect is not instantly gone, but decays exponentially (if the noise follows a Gaussian distribution).
Maths was requested:
The white rounding error std. dev. introduced by digitization is always \$LSB/2/\sqrt3\$. This noise gets added to your signal when digitizing, thus reducing SNR. Digitizing can never enhance SNR - only reduce it. If the noise std. dev. is e.g. \$2 LSB\$ to begin with, you end up with \$ \sqrt{2^2+1/12} = 2.02 LSB\$ of noise, a 1% reduction in SNR. The more noisy bits, the weaker the SNR reduction.
It is a small effect, but as it is completely avoidable - namely by using the smallest full range accomodating the signal+noise - one should be aware of it.
For slow and low resolution converters, the RMS noise in the measured signal can easily be even below 1 LSB when using no pre-gain. In this case, the rounding can constitute the majority of the noise in the digitized signal.