  We generalize the classical Henneberg minimal surface by giving an infinite
family of complete, finitely branched, non-orientable, stable minimal surfaces
in $\mathbb{R}^3$. These surfaces can be grouped into subfamilies depending on
a positive integer (called the complexity), which essentially measures the
number of branch points. The classical Henneberg surface $H_1$ is characterized
as the unique example in the subfamily of the simplest complexity $m=1$, while
for $m\geq 2$ multiparameter families are given. The isometry group of the most
symmetric example $H_m$ with a given complexity $m\in \mathbb{N}$ is either
isomorphic to the dihedral isometry group $D_{2m+2}$ (if $m$ is odd) or to
$D_{m+1}\times \mathbb{Z}_2$ (if $m$ is even). Furthermore, for $m$ even $H_m$
is the unique solution to the Bj\"orling problem for a hypocycloid of $m+1$
cusps (if $m$ is even), while for $m$ odd the conjugate minimal surface $H_m^*$
to $H_m$ is the unique solution to the Bj\"orling problem for a hypocycloid of
$2m+2$ cusps.
