Component importance measures are relevant to improve the system design and to develop optimal replacement policies. Birnbaum's importance measure is one of the most relevant measures. If the components are (stochastically) independent, this measure can be defined using several equivalent expressions. However, in many practical situations, the independence assumption is unrealistic. It also turns out that in the case of dependent components, different Birnbaum's measure definitions lead to different concepts. In this paper, we extend Birnbaum's importance measure to the case of dependent components in a way allowing us to obtain relevant properties including connections and comparisons with other measures proposed and studied recently. The dependence is modeled through copulas and the new measure is based on the contribution of the component to the system reliability.