  Let $X$ be a Hilbert modular variety and $\mathbb{V}$ a non-trivial local
system over $X$ with infinite monodromy. In this paper we study Saito's mixed
Hodge structure (MHS) on the cohomology group $H^k(X,\mathbb{V})$ using the
method of Higgs bundles. Among other results we prove the Eichler-Shimura
isomorphism, give a dimension formula for the Hodge numbers and show that the
mixed Hodge structure is split over $\mathbb{R}$. These results are analogous
to Matsushima-Shimura [Annals of Mathematics 78, 1963] in the cocompact case
and complement the results in Freitag [Book: Hilbert modular forms,
Springer-Verlag, Berlin, 1990] for constant coefficients.
