Additive White Gaussian Noise (AWGN) is typically assumed to be zero mean as this adds random fluctuations around the original signal without biasing it in some way. If there is a non-zero mean in additive noise, it will impart a bias on the signal. This is opposed to Multiplicative White Gaussian Noise, which is typically assumed to be unit mean. This is because multiplicative noise scales the signal, and implies that on average, the signal's scale is not changed. If multiplicative noise is not unit-mean, there will be some bias in the scaling. The signal will on average be amplified if the mean is >1, and on average attenuated if the mean is <1.
For AWGN with assumed zero mean, noise power = noise variance. So, setting the noise to have unit variance normalizes the noise power. In the case where the noise is non unit mean, noise power can be expressed more generally as noise variance plus noise mean squared. If you assume unit variance and add the mean square of the noise, you can calculate the power.