In ancient Greece, people carried a single name, and some names were quite common, while others were very rare. If these names were used to distinguish people, why didn't everybody have a different name? Looking into the manner in which the ancient Greeks picked names, we develop an economic model for the existence of names, as a way of exchanging identification information. Considering different information frameworks, we justify the exchange of names in this context as the best system in order to promote cooperation. We then study the optimal choice of names in these conditions and show that the impact of strategic naming on the distribution of names works as an alteration of existing mean-field approaches to name dynamics, which converge to power laws. Strategic naming adds a degree of freedom in the relationship between the observed number of names and the shape of the power law distribution. Confronting these results to empirical data from the archaic and classical periods, we observe that a form of conformist strategic naming could account for the particular shape of the name distribution in Ancient Greece, which differs from contemporary data.