A partial answer is that a necessary but probably not sufficient condition for ${\cal H}$ is that it needs to have at most $|V|$ hyperedges -- otherwise there can be no edge set $E$ on $V$ such that $\cal H$ is the set of open nbhoods of $G=(V,E)$.
An example of a hypergraph not representable by open nbhoods: Let $V$ be the set $\{1,2,3,4\}$ and ${\cal E}=\{V\setminus\{a,b\}: a<b\in V\}$ and set ${\cal H}=(V,{\cal E})$.