What does G(1) = 1 say about a system?
Static gain refers to the DC gain. Namely, it would be the ratio of the output and the input under steady state condition.

Due to DC corresponding to $\omega=0$, in the $z$-domain DC would correspond to $z=1$ because $z=re^{j\omega}$, with $r=1$ (i.e. you have to evaluate your transfer function $G(z)$ in the unit circle to get the frequency response).

EDIT:

The frequency response of a system is represented in the Fourier domain, $G(\omega)$. The DTFT of a given discrete sequence can be calculated as:

$$G_F(\omega)=\sum_{n=-\infty}^{\infty}g[n]e^{-j\omega n}$$

On the other hand, the $z$-transform is defined as:

$$G_Z(z)=\sum_{n=-\infty}^{\infty}g[n]z^{-n}$$

It's easy to see that if we evaluate $z=e^{j\omega}$ (the unit circle), then the $z$-transform returns the frequency response.