Minimal hypergraph transversals the exact complexity for hypergraph transversal problem is yet unknown and is an open research problem.

However, I would need a fast way to compute, on paper, the minimal transversal of a hypergraph. A transversal intersects all hyperedges of the graph. 

A transversal is minimal if it does not contain any transversal as a proper subset.

Let us look at this example, where V = {a,b,c,d,e}

Hyperedge 1: de
Hyperedge 2: abce
Hyperedge 3: bd


How can I compute, on paper, the minimal transversal of this hypergraph? Computing the minimal transversal with only 2 hyperedges is trivial, but in the 3 hyperedges case, the complexity increases.

Looking forward to your input.
The problem is NP-hard by a reduction from hitting set. Thus, there is no efficient or "easy" algorithm in this sense. Of course, any algorithm you might execute with pen and paper can also be realized on a computer.

With that being said, just try all possibilities by hand. As you observe, it is easy to see if the solution is of size one or two. It can get much more tedious after that, and the reason is NP-hardness.