  We examine unresolved nuclear X-ray sources in 57 brightest cluster galaxies
to study the relationship between nuclear X-ray emission and accretion onto
supermassive black holes (SMBHs). The majority of the clusters in our sample
have prominent X-ray cavities embedded in the surrounding hot atmospheres,
which we use to estimate mean jet power and average accretion rate onto the
SMBHs over the past several hundred Myr. We find that ~50% of the sample have
detectable nuclear X-ray emission. The nuclear X-ray luminosity is correlated
with average accretion rate determined using X-ray cavities, which is
consistent with the hypothesis that nuclear X-ray emission traces ongoing
accretion. The results imply that jets in systems that have experienced recent
AGN outbursts, in the last ~10^7yr, are `on' at least half of the time. Nuclear
X-ray sources become more luminous with respect to the mechanical jet power as
the mean accretion rate rises. We show that nuclear radiation exceeds the jet
power when the mean accretion rate rises above a few percent of the Eddington
rate, where the AGN apparently transitions to a quasar. The nuclear X-ray
emission from three objects (A2052, Hydra A, M84) varies by factors of 2-10 on
timescales of 6 months to 10 years. If variability at this level is a common
phenomenon, it can account for much of the scatter in the relationship between
mean accretion rate and nuclear X-ray luminosity. We find no significant change
in the spectral energy distribution as a function of luminosity in the variable
objects. The relationship between accretion and nuclear X-ray luminosity is
consistent with emission from either a jet, an ADAF, or a combination of the
two, although other origins are possible. We also consider the longstanding
problem of whether jets are powered by the accretion of cold circumnuclear gas
or nearly spherical inflows of hot keV gas.[abridged]

Cosmology and Nongalactic Astrophysics