There's one contingincy you haven't calculated. What is the probability if we don't know what he would have done. In other words, what is the answer to the question as it was put?There's one big issue at the heart of it (just like in the Monty Hall problem), JeffJo alluded to it, and it is the one that decides whether the answer is 13/27, 1/3 or 1/2.
If your friend did not have a boy born on Tuesday, what would he have done instead?
If he would've picked one kid at random and asked the question anyway, using that kid's data, the answer is 1/2.
If he would've stayed silent, the answer is 13/27.