The Metabolically Coupled Replicator System (MCRS) model of early chemical evolution offers a plausible and efficient mechanism for the self-assembly and the maintenance of prebiotic RNA replicator communities, the likely predecessors of all life forms on Earth. The MCRS can keep different replicator species together due to their mandatory metabolic cooperation and limited mobility on mineral surfaces, catalysing reaction steps of a coherent reaction network that produces their own monomers from externally supplied compounds. The complexity of the MCRS chemical engine can be increased by assuming that each replicator species may catalyse more than a single reaction of metabolism, with different catalytic activities of the same RNA sequence being in a trade-off relation: one catalytic activity of a promiscuous ribozyme can increase only at the expense of the others on the same RNA strand. Using extensive spatially explicit computer simulations we have studied the possibility and the conditions of evolving ribozyme promiscuity in an initial community of single-activity replicators attached to a 2D surface, assuming an additional trade-off between replicability and catalytic activity. We conclude that our promiscuous replicators evolve under weak catalytic trade-off, relatively strong activity/replicability trade-off and low surface mobility of the replicators and the metabolites they produce, whereas catalytic specialists benefit from very strong catalytic trade-off, weak activity/replicability trade-off and high mobility. We argue that the combination of conditions for evolving promiscuity are more probable to occur for surface-bound RNA replicators, suggesting that catalytic promiscuity may have been a significant factor in the diversification of prebiotic metabolic reaction networks.