Dude, I just burned an hour trying to understand the Tuesday Boy problem since I've never heard of it until now. I have two contributions.
1) It's best understood spatially. There's a hit on Google that emphasizes that although two different boys can both be born on Tuesday, the "hit" only counts for one in terms of odds since they must co exist. That's why the answer is close to 50% but not quite 50%.
2) Implicitly distributable attribution. If I say I have a random person here of unknown gender, you immediately think, hey, 50/50 male/female (apologies to the LGBT community). In the question, you read born on a Tuesday. You could interpret the date as meaningless or implicitly distributable. If it had the same social gravity as gender, you could easily see how the solution above applies. But if you think date has nothing to do with birth rates but somehow isn't random, then the answer could very well be anything. 0, 100, 73, i.e. there may be a date (Friday!) where mad babies are born.
It's really hard to chew through at first but those two nuggets have comforted me in settling down on the Internet's correct answer of 13/27.