Variance of N samples of Noise
$\sigma^2$ of the sampled version is $\sigma^2$ except for a possible scaling of units (such as volts-squared to counts-squared)

When you sample the signal the variance does not change until subsequent digital processing such as filtering is done (assuming the sampling rate and the signal itself are uncorrelated). 

Consider the simple case of a sine wave; in the analog the variance is $V_p^2/2$ where $V_p$ is the peak amplitude. If we sample the sinewave for a long enough time with an incommensurate (not an integer multiple or sub-multiple of the sine wave) sampling clock, we can see without much trouble that the statistical distribution characteristics of the sine wave (histogram, variance, standard deviation etc) are unchanged.

Further, what is very interesting is if we sample a signal with a spectral power distribution that goes beyond our sample rate (and we did not implement any sort of anti-alias filter that would have reduced the overall variance), the variance of the sampled signal (again if the sampling clock is not correlated to the signal) is unchanged!  This again is very clear by imagining the histogram of the analog and digital signals; in both cases analog and digital, with enough samples from the analog signal the histograms would have the same Gaussian shape and width (standard deviation) and variance!

This is perhaps made clearer from the plot below of samples of a  Gaussian distributed waveform. Regardless of the sampling rate, as long as we sample long enough to establish a reasonable statistical distribution, we will still result in the same Gaussian shape with the same standard deviation.


It is because of this reason that I take great precautions in making spectral density estimations from digital samples, to ensure that the higher Nyquist zones have been properly filtered prior to sampling. Once digital, without knowledge of the anti-alias filtering that was done, there is no way to tell if the spectral noise density is identical to the analog at a particular frequency offset, or the power summation of multiple Nyquist zones that have aliased into the band! However (and interesting!) if I am only interested in the total power (variance) of an analog signal, regardless of frequency location of that power (spectral density), then I can sample that signal at virtually any rate (again important to know that the sampling clock and the signal are not correlated, which for AWGN as in your case they would not be) and measure the variance of the digital signal and no that it is equal to the variance of the analog signal. As might now be clear, if the digital sampling process low pass filters the analog signal (which in practice for a "white" noise signal it absolutely will!), then the digital variance will be less than the analog. To avoid this, we would make sure that the analog signal is first low pass filtered to below the analog bandwidth of the digital sampling system (which may and often is much higher than the sampling rate).