"At dinner party for adults only, you find out that Mrs. Smith has exactly two children of different ages. You also find out that she was a Cub Scout Den Mother when they were younger. What are the chances she has two boys?"
The key to this version, is that the property "at least one boy" is identified as a property of the two children as a pair, not of one specific child in the pair. If it was one specific child, no matter how or who selected him or her, the answer can only be 1/2.
I know that this can be hard to accept. So I'll demonstrate with a variation of what is currently the top answer, Yamikuronue's. Note that I'm not singling that one out for any reason other than it is at the top.
"At a dinner party the other night I spoke with Mr Smith. He mentioned having two kids. He also told a story about one of them attending a special event for children of only one gender, although he didn't mention whether it was for boys or girls. What are the chances that he has two children of the same gender?"
If Mr. Smith had mentioned that it was a boys' event, the question is the same as Yamikuronue's with some details filled in about the story. If he had mentioned that it was a girls' event, it is an equivalent question that can only have the same answer. Since every possibility produces the same answer, I don't need to know what gender the event was for in order to give that answer.
But I also don't have any information about Mr. Smith's children. With two children and no information about either child's gender, the chances that he has two of the same gender are 1/2. Yet, if the answer to Yamikuronue's question could be 1/3, we'd have a paradox since a story which provides no useful information changes the chances from 1/2 to 1/3.
This paradox has a name: Bertrand's Box Paradox. While some modern treatments use that name for the problem, Bertrand used it to show that you have to know how this type of information is obtained to change the answer. And in fact, the Two Child Problem is essentially Bertrand's Box Problem, with an added box containing one of each kind of coin/gender.