We examine a naming game on an adaptive weighted network. A weight of connection for a given pair of agents depends on their communication success rate and determines the probability with which the agents communicate. In some cases, depending on the parameters of the model, the preference toward successfully communicating agents is essentially negligible and the model behaves similarly to the naming game on a complete graph. In particular, it quickly reaches a single-language state, albeit some details of the dynamics are different from the complete-graph version. In some other cases, the preference toward successfully communicating agents becomes much more important and the model gets trapped in a multi-language regime. In this case gradual coarsening and extinction of languages lead to the emergence of a dominant language, albeit with some other languages still present. A comparison of distribution of languages in our model and in the human population is discussed.