In this answer we will basically expand on Lubos Motl's correct answer using some other words and introducing some terminology.
In the Hermitian one-matrix model, the action
$$\tag{1}S~=~ {\rm Tr} L(H)$$
is invariant under adjoint conjugation
$$\tag{2} H\to UHU^{-1}$$
with unitary matrices $U$. Eq.(2) here play the role of the gauge transformations. One therefore has $N^2$ real gauge parameters.
At the infinitesimal level $U=e^A$, the gauge transformation is
$$\tag{3} \delta H ~=~[A, H],$$
where $A$ is an infinitesimal anti-Hermitian matrix.
On the other hand, the $N$ real eigenvalues $\lambda_1$, $\ldots$, $\lambda_N$ of an Hermitian matrix are gauge invariants, which cannot be changes by gauge transformations of adjoint type (2). Hence there are actually only $N(Nā1)$ independent gauge parameters. Such a gauge algebra is called reducible.
For a diagonal Hermitian matrix $H$, the $N$ redundant gauge parameters may be identified with the diagonal matrix entries $A^1{}_1$ , $A^2{}_2$, $\ldots$, $A^N{}_N$, at the infinitesimal level.
The Faddeev-Popov trick (in its original formulation) applies to irreducible gauge symmetries, but one can make it work in this reducible case (i.e. the Hermitian one-matrix model) by properly identifying the independent gauge parameters, cf. above.
The Faddeev-Popov determinant becomes the square of the Vandermonde determinant of the eigenvalues $\lambda_1$, $\ldots$, $\lambda_N$.
Finally, it seems natural to mention that the presence of this Vandermonde determinant is a typical feature of (random) matrix models, and it leads to eigenvalue repulsion, cf. e.g. this Phys.SE post.
