Longitudinal data of conventionalization in emerging languages, combined with computational models explaining such data, are lacking in the literature on language emergence. In the present study we report on the emergence of gestural communication systems (“homesigns”) invented by deaf individuals in Nicaragua. Analysis of longitudinal data from several families shows gradual convergence toward a gestural system with the essential characteristics of a shared lexicon. We propose a general computational framework to formalize the linguistic and social interactions among the individual signers such that a shared lexicon may arise. More specifically, a reinforcement learning process that adjusts the individual’s probability of gesture use in response to others’ actual gesture use provides a suitable account of the observed gestural convergence. Implications for language emergence are discussed.