How to derive the Method of Moments estimator of mu, using the second moment of X, when X is norm distributed? If $X_1,\ldots,X_n$ follow a normal distribution, where the variance $\sigma$ is given, how can you derive the MME of the mean $\mu$ using the second moment?

Note that the second moment of $X$ is
$$E[X^2] = \sigma^2 + \mu^2,$$
so by subtracting the known variance from the second moment, you can find the square of the mean.
To this end, compute an estimate of the second moment
$$\hat s = \frac {1}{N} \sum_{i=1}^N X_i^2,$$
then find the estimate of the mean with
$$\hat \mu = \sqrt{\hat s -\sigma^2}.$$
