If the $x_j$'s are distinct modulo $1$ (which is the natural assumption), then
$$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\left|\sum_{j}c_j e^{2\pi i n x_j}\right|^2=\sum_{j}|c_j|^2.$$
Indeed,
$$\sum_{n=1}^{N}\left|\sum_{j}c_j e^{2\pi i n x_j}\right|^2=\sum_{j,k}c_j\overline{c_k}\sum_{n=1}^N e^{2\pi in(x_j-x_k)}.$$
The inner sum equals $N$ for $j=k$, and it is bounded by $|\sin\pi(x_j-x_k)|^{-1}$ in absolute value for $j\neq k$. The result follows.