It depends on how the "given" information is determined:
You say "Gee, Mr. Smith, I hear you have two children. Is either, or both, perhaps a boy who was born on a Tuesday?" Mr. Smith replies "That's an odd question. The chances are only 27/196 that you would have guessed right, but in this case you did. At least one of my children is a boy who was born on a Tuesday."
You say "Gee, Mr. Jones, I hear you have two children. Can you tell me an odd fact that applies to at least one of them?" Mr. Jones replies "Well, at least one is a boy who was born on a Tuesday."
The answer in case #1 is 13/27. There are 196 possible combinations of day+gender in a two-child family. 27 of those include Tuesday Boy, and 13 of those have two boys.
The answer in case #2 is 1/2, because Mr. Jones could have told you about, say, a girl who was born on a Thursday when he also had a boy who was born on a Tuesday. In fact, of the 27 combinations, 26 include a different form of the day+gender description. If you didn't ask about Tuesday Boys, you can only assume Mr. Jones would choose randomly between the two. So you have to dismiss 13 cases where he has a Tuesday Boy, including 6 where he has two. The answer is (13-6)/(27-13)=7/14=1/2.
What seems to perplex far too many people, is that the requirement that we know "one is a boy who was born on a Tuesday" is not the same event as the observation about one child, that he "is a boy who was born on a Tuesday."