What is a hypergraph minor? Is there a theory of hypergraph minors? I could only find some attempts to define them at papers/theses, whose main topic was something else. What would be a useful definition? Does the hypergraph version of the Robertson–Seymour theorem hold?
 A: I think both questions of the OP are answered in: Hypertree-depth and minors in hypergraphs (2012)

Let $H$ and $H′$ be hypergraphs. Then $H$ is a minor of $H′$ if $H$
  can be obtained from $H′$ by a sequence of operations of the following
  kinds:
  • vertex deletion,
  • contraction of (the edge between) two vertices that are contained in a common hyperedge,
  • addition of ahyperedge $e$ such that the set $e$ induces a clique in the underlying graph, and
  • deletion of a proper subhyperedge.
For graphs, a famous result by Robertson and Seymour shows that
  testing for a fixed minor is solvable in cubic time. In contrast,
  testing for a fixed hypergraph minor can be NP-hard (section 6.4).

A: In a note concerning a version of Hadwiger's conjecture for hypergraphs, I once defined what a (complete) minor map of a hypergraph is, and probably this can be used to define a notion of "minor of hypergraph". (If you are interested in the note, which I did not publish or put on arXiv, I am happy to send it to you.)
For the following definitions assume that $H=(V,E)$ is a hypergraph.
A subset $\emptyset \neq S \subseteq V$ is said to 
be connected if for all $S_1 \subseteq S$ with $\emptyset \neq S_1
\neq S$ there are $x\in S_1, y\in S\setminus S_1$ and $a \in E$ such 
that $x,y \in a$.
Let $S,T \neq \emptyset$ be disjoint subsets of $V$. 
They are said to be
connected to each other if there are $s \in S, t\in T$ and $a\in E$
such that $x,y \in a$.
A map $m: \kappa \to {\cal P}(V)\setminus \{\emptyset\}$
is said to be a complete minor map  if


*

*$m(x)$ is connected for all $x\in \kappa$;

*for $x\neq y \in \kappa$ we have $m(x)\cap m(y) = \emptyset$ and
$m(x),m(y)$ are connected to each other.

