Countably generated sigma-algebras

Claim: Under CH, there is no such family.

Proof: Recall that $\mathcal{P}(\omega_1) \otimes \mathcal{P}(\omega_1) = \mathcal{P}(\omega_1 \times \omega_1)$ (a result of B. V. Rao, On discrete Borel spaces and projective sets, Bull Amer. Math. Soc. 75 (1969), 614–617). Given $\{A_i : i < \omega_1\} \in [\omega_1]^{\omega_1}$, choose $\{(X_n, Y_n): n < \omega\}$, $X_n, Y_n \subseteq \omega_1$, such that the sigma algebra generated by $\{X_n \times Y_n : n < \omega \}$ contains $\{(i, x): i < \omega_1, x \in A_i\}$. It follows that the sigma algebra generated by $\{Y_n : n < \omega\}$ contains each $A_i$.

In the other direction, Arnold W. Miller, Generic Souslin sets, Pacific J. Math. 97 (1981), 171–181 showed that it is consistent that there is no countable family $\cal{F} \subseteq \mathcal{P}(\mathbb{R})$ such that the sigma algebra generated by $\mathcal{F}$ contains all analytic sets. So the existence of such a family is independent of ZFC.