I am wondering if there is a general explanation for the following phenomenon. The partial sums of the geometric series $\sum_{n\geq 0} x^n$ evaluated at a root of unity $\zeta\neq 1$ in the complex plane attain only finitely many values (since $1+\zeta+\ldots+\zeta^{n1}=0$, where $\zeta^n=1$). The average of these values is $1/(1\zeta)$, which is the value at $\zeta$ of the meromorphic function that analytically continues the geometric series. Out of curiosity, I would like to ask if this is a special case of a general theorem?

The phenomenon you observe is a special case of a theorem of Frobenius (1880):
In your case the series is $(1,\zeta,\zeta^2,\dots)$ which ensures Cesaro summability as the sequence of partial sums is periodic. For more details see Page 4 of Korevaar: Tauberian theory  A century of developments (Springer, 2004). 

