We present a model for the distribution of family names that explains the power-law decay of the probability distribution for the number of people with a given family name. The model includes a description of the process of generation or importation of new names, and a description of the growth of the number of individuals with a name, and corresponds, for a long-enduring culture, to a Galton-Watson branching process killed at a random time. The exponent that characterizes the decay of the resulting distribution is determined by the characteristic rates for the creation of new names and for the growth of the population. The power-law decay is modulated by small-amplitude log-periodic oscillations. This is rigorously established for a particular form of the offspring distribution in the branching process, but arguments are presented to show that the phenomenon will occur under wide circumstances.