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]
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