To my knowledge, Chomsky does not claim that there is a specific gene or mutation responsible for the human linguistic abilities - at least not in strictly biological sense. The claims of Chomsky and his followers, known as generative grammar, are more general: the study and the comparison of many human languages have shown that all the languages are structured according to the same principles (as opposed to the earlier Sapir-Whorf hypothesis that languages are products of purely random evolution). This similarity is the basis for the Chomsky's claim that this language structure is likely hard-coded in the human brain. This is further supported by mathematical complexity arguments, showing that the language rules are too complex, and the variety of the possible phrases that one can produce is too large to be learned within a reasonable time, without having a prior model of how the langauge is structured.
Thus, the linguistic ability, like many other traits, is likely a result of many mutations and action (and interaction) of many genes. Note that this involves not only the mental abilities, but also purely physical capacity to produce wide range of sounds that many animals do not possess.
Remark: Noam Chomsky is a renowned mathematical linguist, chose theories of generative/transformational grammar are pretty much the basis of the modern mainstream linguistics, and play an important role in the modern computer science. Thus, these are not to be dismissed, unlike his more controversial political views, which are anything but mainstream.