Assume I have a set $H_k$ of all weakly connected directed graphs with $k$ unlabeled vertices. I want to find a directed graph $G$ such that each graph $H \in H_k$ is an induced subgraph of $G$. Furthermore, what is the minimum number of vertices that $G$ could have?
The trivial solution is of course a graph that is the union of all subgraphs which would be a upper bound of $k * N(H_k)$ vertices where $N(H_k)$ is the size of $H_k$. A lower bound could be found by finding some number $n$ such that $\binom{n}{k} < N(H_k) \le \binom{n+1}{k}$ as $\binom{n}{k}$ would be the number of subgraphs of size $k$ a complete graph with $n$ vertices can possibly contain. Not sure how to drill this down further though.
Is there a name for this kind of problem of creating a minimum graph that contains a set of subgraphs?
This is called a [induced] universal [di]graph for the family of all weakly connected digraphs with k vertices.
Tight bounds are known for the family of all digraphs on $k$ vertices (regardless of connectivity): at least $2^{k-1}$ vertices are necessary and not much more are known to be sufficient, $2^{k-1}(1+o(1))$. See section 4.2 of Noga Alon's Asymptotically optimal induced universal graphs and references there for simpler techniques that prove that $16 \cdot 2^{k-1}$ vertices suffice.
The same upper bounds of course apply to weakly connected digraphs too. And perhaps the lower bound applies as well (I did not check).
One frequent idea, more of a starting point really, is noticing the equivalence to adjacency labeling schemes: a universal graph with $N$ vertices for a family $F$ gives you a way to label vertices of any graph in $F$ with labels in $\{1,\dots,N\}$ so that adjacency between any two vertices is determined by those labels alone. This works in the other direction too. So you can try to develop clever ways of encoding adjacency information into vertex labels, minimizing the number of distinct labels (e.g. minimising the number of bits, if labels are binary strings).
There's quite a few presentations on youtube if you search "Universal graph labeling scheme" (though mostly for undirected graphs and restricted families such as planar graphs).