The distributed naming problem, assigning different names to the nodes in a distributed system, is a fundamental task. This problem is non trivial, specially when the amount of memory available for the task is low, and when requirements for fault-tolerance are added. The considered distributed computation model is population protocols. In this model, a priori anonymous and indistinguishable mobile nodes (called agents), communicate in pairs and in an asynchronous manner (according to a fairness condition). Fault-tolerance is addressed through self-stabilization, in terms of arbitrary initialization of agents. This work comprises a comprehensive study on the necessary and sufficient state space conditions for naming. It is studied under various combinations of model assumptions: weak or global fairness, arbitrary or uniform initialization of agents, existence or absence of a distinguishable agent (arbitrarily initialized or not), possibility of breaking symmetry in pair-wise interactions (symmetric or asymmetric transitions). For each possible combination of these assumptions, either an impossibility is proven or the necessary exact number of states (per mobile agent) is determined and an appropriate space-optimal naming protocol is presented.