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Jiro comments on Bayes' Theorem Illustrated (My Way) - Less Wrong
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This sounds like you are trying to divide "two boys born on Tuesday" into "two boys born on Tuesday and the person is talking about the first boy" and "two boys born on Tuesday and the person is talking about the second boy".
That doesn't work because you are now no longer dealing with cases of equal probability. "Boy 1 Monday/Boy 2 Tuesday", "Boy 1 Tuesday/Boy 2 Tuesday", and "Boy 1 Tuesday/Boy 1 Monday" all have equal probability. If you're creating separate cases depending on which of the boys is being referred to, the first and third of those don't divide into separate cases but the second one does divide into separate cases, each with half the probability of the first and third.
As I pointed out above, whether it adds information (and whether the analysis is correct) depends on exactly what you mean by "one is a boy born on Tuesday". If you picked "boy" and "Tuesday" at random first, and then noticed that one child met that description, that rules out cases where no child happened to meet the description. If you picked a child first and then noticed he was a boy born on a Tuesday, but if it was a girl born on a Monday you would have said "one is a girl born on a Monday", you are correct that no information is provided.
The only relevant information is that one of the children is a boy. There is still a 50% chance the second child is a boy and a 50% chance that the second child is a girl. Since you already know that one of the children is a boy, the posterior probability that they are both boys is 50%.
Rephrase it this way:
I have flipped two coins. One of the coins came up heads. What is the probability that both are heads?
Now, to see why Tuesday is irrelevant, I'll re-state it thusly:
I have flipped two coins. One I flipped on a Tuesday and it came up heads. What is the probability that both are heads?
The sex of one child has no influence on the sex of the other child, nor does the day on which either child was born influence the day any other child was born. There is a 1/7 chance that child 1 was born on each day of the week, and there is a 1/7 chance that child 2 was born on each day of the week. There is a 1/49 chance that both children will be born on any given day (1/7*1/7), for a 7/49 or 1/7 chance that both children will be born on the same day. That's your missing 1/7 chance that gets removed inappropriately from the Tuesday/Tuesday scenario.
No there's not. The cases where the second child is a boy and the second child is a girl are not equal probability.
If you picked "heads" before flipping the coins, then the probability is 1/3. There are three possibilities: HT, TH, and HH, and all of these possibilities are equally likely.
If you picked "heads" and "Tuesday" before knowing when you would be flipping the coins, and then flipped each coin on a randomly-selected day, and you just stopped if there weren't any heads on Tuesday, then the answer is the same as the answer for boys on Tuesday. If you flipped the coin and then realized it was Tuesday, the Tuesday doesn't affect the result.
If you picked the sex first before looking at the children, the sex of one child does influence the sex of the other child because it affects whether you would continue or say "there aren't any of the sex I picked" and the sexes in the cases where you would continue are not equally distributed.