  In this article we introduce a definition for the moduli space of equivariant
minimal immersions of the Poincar\'e disc into a non-compact symmetric space,
where the equivariance is with respect to representations of the fundamental
group of a compact Riemann surface of genus at least two. We then study this
moduli space for the non-compact symmetric space $\mathbb{RH}^n$ and show how
$SO_0(n,1)$-Higgs bundles can be used to parametrise this space, making clear
how the classical invariants (induced metric and second fundamental form)
figure in this picture. We use this parametrisation to provide details of the
moduli spaces for $\mathbb{RH}^3$ and $\mathbb{RH}^4$, and relate their
structure to the structure of the corresponding Higgs bundle moduli spaces.
