  If $Y$ is a properly embedded minimal surface in a convex cocompact
hyperbolic 3-manifold $M$ with boundary at infinity an embedded curve $\gamma$,
then Graham and Witten showed how to define a renormalized area $\calA$ of $Y$
via Hadamard regularization. We study renormalized area as a functional on the
space of all such minimal surfaces. This requires a closer examination of these
moduli spaces; following White and Coskunuzer, we prove these are Banach
manifolds and that the natural map taking $Y$ to $\gamma$ is Fredholm of index
zero and proper, which leads to the existence of a $\ZZ$-valued degree theory
for this mapping. We show that $\calA(Y)$ can be expressed as a sum of the
Euler characteristic of $Y$ and the total integral of norm squared of the
trace-free second fundamental form of $Y$. An extension of renormalized area to
a wider class of nonminimal surfaces has a similar formula also involving the
integral of mean curvature squared. We prove a formula for the first variation
of renormalized area, and characterize the critical points when $M = \HH^3$ and
$\gamma$ has a single component. All of these results have analogues for
4-dimensional Poincar\'e-Einstein metrics. We conclude by discussing the
relationship of $\calA$ to the Willmore functional.
