We have surveyed the ~0.1-10 MeV nucleon-1 abundances of heavy ions from 3He through Fe in 64 large solar energetic particle (LSEP) events observed on board the Advanced Composition Explorer from 1997 November through 2005 January. Our main results are (1) the 0.5-2.0 MeV nucleon-1 3He/ 4He ratio is enhanced between factors of ~2-150 over the solar wind value in 29 (~46%) events. (2) The Fe/O ratio in most LSEP events decreases with increasing energy up to ~60 MeV nucleon -1. (3) The Fe/O ratio is independent of CME speed, flare longitude, event size, the 3He/4He ratio, the pre-event Fe/O ratio, and solar activity. (4) The LSEP abundances exhibit unsystematic behavior as a function of M/Q ratio when compared with average solar wind values. (5) The survey-averaged abundances are enhanced with increasing M/Q ratio when compared with quiet coronal values and with average gradual SEP abundances obtained at 5-12 MeV nucleon-1. (6) The event-to-event variations in LSEP events are remarkably similar to those seen in CME-driven IP shocks and in 3He-rich SEP events. The above results cannot be explained by simply invoking the current paradigm for large gradual SEP events, i.e., that CME-driven shocks accelerate a seed population dominated by ambient coronal or solar wind ions. Instead, we suggest that the systematic M/Q-dependent enhancements in LSEP events are an inherent property of a highly variable suprathermal seed population, most of which is accelerated by mechanisms that produce heavy-ion abundances similar to those observed in impulsive SEP events. This heavy-ion-enriched material is subsequently accelerated at CME-driven shocks near the Sun by processes in which ions with higher M/Q ratios are accelerated less efficiently, thus causing the Fe/O ratios to decrease with increasing energy.