Is there a rule of thumb for selecting the variance of the input?
You can define your model as $$\hat{x}[n] = \frac{1}{||\mathbf{x}_{n}||}\mathbf{A}^T\mathbf{x}_{n} +\frac{1}{\sqrt{\mathsf{SNR}}} w[n]$$
In this way, you define a range for $\mathsf{SNR}$ in dB (e.g. between 0 to 10 dB), then translate it to natural (non-dB) scale, and then $\frac{1}{\sqrt{\mathsf{SNR}}}$ gives you the noise power. The signal power is normalized to one which makes it easy.

Here, $w\sim \mathcal{N}(0,1)$ is a standard Gaussian RV and you can simulate it as w = randn;. In fact, $\frac{1}{\sqrt{\mathsf{SNR}}}w=w'\sim \mathcal{N}(0,\sigma^2_{w'})$, where $\sigma^2_{w'}$ has the right proportion wrt the SNR.