You can show that every $\sigma$-algebra on a finite set is defined by a partition (and it is isomorphic to $\mathcal P(S)$, where $S$ is the defining partition).
The way you do it is by showing that if $B$ is a $\sigma$-algebra on $X$, then below every non-zero element, there is a minimal positive element (or an atom in the technical jargon). You can then show that the set of atoms is a partition, and that there is a canonical isomorphism between $B$ and $\mathcal P(S)$, where $S$ is the set of atoms.
So the question is how many partitions a finite set has. And the answer is not pretty in the form of Bell numbers.
So in order to count how many $\sigma$-algebras contain a certain family of subsets, you need to ask yourself what partitions can be used to generate the family of sets. I am not aware of any "pretty" combinatorial term for this, but it can't be pretty.