I'm surprised that no one has mentioned the differential-geometric motivation for the Casimir element.

Suppose $\mathfrak{g}$ is the Lie algebra of a compact Lie group $G$.  Then $U(\mathfrak{g})$ is the algebra of left-invariant differential operators on $G$. 

$G$ has a natural bi-invariant metric--given by the Killing form.  But now there is an obvious central element of $U(\mathfrak{g})$: the Laplacian associated to this metric!  This is precisely the Casimir element; I had always assumed this was the original motivation for the construction.