  We compare accretion and black hole spin as potential energy sources for
outbursts from AGN in brightest cluster galaxies (BCGs). Based on our adopted
spin model, we find that the distribution of AGN power estimated from X-ray
cavities is consistent with a broad range of both spin parameter and accretion
rate. Sufficient quantities of molecular gas are available in most BCGs to
power their AGN by accretion alone. However, we find no correlation between AGN
power and molecular gas mass over the range of jet power considered here. For a
given AGN power, the BCG's gas mass and accretion efficiency, defined as the
fraction of the available cold molecular gas that is required to power the AGN,
both vary by more than two orders of magnitude. Most of the molecular gas in
BCGs is apparently consumed by star formation or is driven out of the nucleus
by the AGN before it reaches the nuclear black hole. Bondi accretion from hot
atmospheres is generally unable to fuel powerful AGN, unless their black holes
are more massive than their bulge luminosities imply. We identify several
powerful AGN that reside in relatively gas-poor galaxies, indicating an
unusually efficient mode of accretion, or that their AGN are powered by another
mechanism. If these systems are powered primarily by black hole spin, rather
than by accretion, spin must also be tapped efficiently in some systems, i.e.,
$P_{\rm jet} > \dot Mc^2$, or their black hole masses must be substantially
larger than the values implied by their bulge luminosities. We constrain the
(model dependent) accretion rate at the transition from radiatively inefficient
to radiatively efficient accretion flows to be a few percent of the Eddington
rate, a value that is consistent with other estimates.
