I believe, but can't prove, that human language evolved from a combination of gesture and innate vocalizations, via the concomitant evolution of mirror neurons, and that birds will provide the best model for language evolution.
Work on mirror neurons over the past decade has provided intriguing evidence, although no solid proof, for the gestural origins of speech. What can be called the mirror neuron hypothesis(MNH) suggests that only a small re-organization of the nonhuman primate brain was needed to create the wiring that underlies speech acquisition/learning. What is missing from the MNH is a model of the development of language from speech; it is here that I believe that a model based on avian vocalizations is most valuable.
First, some background. Passerine birds can be divided into two groups: the oscines, who learn their songs, and the sub-oscines, who have a limited number of what seem to be innately-specified songs; the former have a well-defined neural architectures and mechanisms for song acquisition; the latter lack brain structures for song acquisition, although they obviously have brain and vocal tract structures for producing song. The sub-oscines, in parallel with nonhuman primates, often use various activities or gestures (posture, numbers of repetitions of songs, feather erectness, types of flights, etc) to provide additional information about the meaning of their utterances. W. John Smith, for example, can predict a flycatchers actions by the combination of posture, flight, and singing pattern he observes. The songbirds, like human children learning language, will not learn their vocalizations if deafened, and need to hear, babble and practice songs before attaining adult competence; very recent work by Rose et al. demonstrate that even the syntax of their song is learned through early exposure to paired phrases, which are then combined to create the adult vocalizations. Such data, demonstrating how sparrows integrate information about temporally-related events and how they use that information to develop sequential vocal behavior, is a viable model for human syntax acquisition.
Now, no one knows if any birds have any mirror neurons, and how their mirror neurons would function if they did exist; some neural data on responses to self-song provide intriguing hints but go no further. I predict (a) the existence of such neurons in oscines and (b) that such neurons will have a robust role in oscine song development, but (c) that only more primitively-functioning mirror neurons (akin to the differences separating monkey and human MNs) will be found in sub-oscines.
Now, what about the so-called missing link between learned and unlearned vocal behavior? No one has found such a missing link in the primate line. But Donald Kroodsma has recently discovered a flycatcher (a supposedly sub-oscine bird) that apparently learns its song. The song is simple, but has variations among groups of birds that constitute dialects. No one yet knows if these birds have brain mechanisms for song learning, or what these mechanisms might be. But I predict that Kroodsma's flycatchers will have mirror neurons that function in intermediate manner, between those of the oscines and sub-oscines, and will provide a model for the missing link between nonhuman primate and human communication.