To evaluate this probability we would need to know the conditional probabilities of M. Smith saying this, given the various cases. Since this information is not available, we can only speculate. As usual in such questions (but not in real life), we assume that the a priori probabilities are $1/4$ that both children would be girls, $1/2$ that one is a girl and one a boy, $1/4$ that both are boys. Here are three scenarios:
A) M. Smith is choosing at random one of his children, and telling you that child's name. There is nothing special about the name "Jane".
B) Without knowing anything about Smith's family, you asked him "How many children do you have? Is one named Jane?"
C) Without knowing anything about Smith's family, you asked him "How many children do you have? Tell me the name of one of your girls, if you have any." There is nothing special about the name "Jane".
In case (A), the probability that both are girls is 1/2 (because in the boy-girl case he might have chosen to tell you about a boy).
In case (B), the probability is approximately 1/2, because the probability of a Jane in a two-girl family is approximately twice the probability of a Jane in a one-girl family (this will depend on Smith's naming practices, but in typical models it will be exactly twice).
In case (C), the probability that both are girls is 1/3, because the conditional probability of the given response is the same in the boy-girl case as in the girl-girl case.