The problem is NP-complete. This rules out an exact algorithm that works in all circumstances, but does not rule out heuristic algorithms that work well in practice, or approximations algorithm with provable approximation guarantees.
The reduction is from 3SAT. Given a 3SAT instance $\phi$ with variables $x_1,\ldots,x_n$ and clauses $\phi_1,\ldots,\phi_m$, construct the following set system. For each variable $x_i$ there are two sets $A_{i,0}$ and $A_{i,1}$ and $N=n+1$ sets $B_{i,t} = \{\beta_{i,t,0},\beta_{i,t,1}\}$, and for each clause $\phi_j$ there is a set $C_j = \{\gamma_{j,1},\gamma_{j,2},\gamma_{j,3}\}$. The set $A_{i,b}$ consists of the following elements:
- The $N+1$ elements $\alpha_i,\beta_{i,1,b},\ldots,\beta_{i,N,b}$.
- For each clause $\phi_j$ that contains $x_i$ as the $k$th literal and is not satisfied by $x_i = b$, the element $\gamma_{j,k}$.
One can find $n(N+1)+m$ diverse sets if and only if $\phi$ is satisfiable. Indeed, given a satisfying assignment $\vec{x}$, the family $\{ A_i^{x_i} : i \in [n] \} \cup \{ B_{i,t} : i \in [n], t \in [N] \} \cup \{ C_j : j \in [m] \}$ is diverse: $\alpha_i$ belongs only to $A_i^{x_i}$, $\beta_{i,t,1-x_i}$ belongs only to $B_{i,t}$, and if the $k$th literal of $\phi_j$ is satisfied then $\gamma_{j,k}$ belongs only to $C_j$.
For the converse, suppose $\mathcal{S} = \mathcal{A} \cup \mathcal{B} \cup \mathcal{C}$ is a diverse family of size at least $n(N+1)+m$, partitioned according to the type of the set. If $\mathcal{A}$ contains both $A_{i,0}$ and $A_{i,1}$ for some $i$, then $B_{i,1},\ldots,B_{i,N} \notin \mathcal{B}$. Hence $|\mathcal{S}| \leq 2n + (n-1)N + m < n(N+1) + m$, which is impossible. Therefore $\mathcal{B}$ and $\mathcal{C}$ must contain all sets of the corresponding type, and $\mathcal{A}$ must contain $n$ sets, which together encode an assignment $\vec{x}$. Since $C_j \in \mathcal{S}$ is diverse, by construction the assignment $\vec{x}$ satisfies clause $\phi_j$, hence $\phi$ is satisfiable.
