I say the probability is 1/2, if we forget all biological and social effects (twins, different chances of having boys and girls, effects of diet on sex of children, whatever).
The reasoning above is almost good. There are 27 cases, 13 in which the other child is a boy, 14 in which the other child is a girl. But I still claim the probability is 1/2.
Why ? Because the formula 13/27 only works if there is equi-probability between the 27 cases, and I think the case of two boys born on a Tuesday is twice as likely as the other 26 cases. For a simple reason : the question would be twice as likely to be asked this way in that case.
Let's consider the broader picture. With no additional knowledge on why the question is asked on one child or the other, we can consider that we have a quintuplet of (child1 sex, child1 day, child2 sex, child2 day, child referred to in the first part of the question).
That gives 14*14*2 possibilities, like :
(boy, Monday, boy, Monday, child1)
(boy, Monday, boy, Tuesday, child1)
(boy, Monday, boy, Wednesday, child1) ...
(boy, Monday, girl, Saturday, child1)
(boy, Monday, girl, Sunday, child1)
(boy, Tuesday, boy, Monday, child1)
(boy, Tuesday, boy, Tuesday, child1) ...
(girl, Sunday, girl, Saturday, child1)
(girl, Sunday, girl, Sunday, child1)
And then exactly the same with the question being posed on child 2 :
(boy, Monday, boy, Monday, child2) ...
(girl, Sunday, girl, Sunday, child2)
If you filter to all matching cases, that is, the cases on which the question refers to boy born on a Tuesday, it gives :
(boy, Tuesday, boy, Monday, child1)
(boy, Tuesday, boy, Tuesday, child1)
(boy, Tuesday, boy, Wednesday, child1) ...
(boy, Tuesday, girl, Saturday, child1)
(boy, Tuesday, girl, Sunday, child1)
(boy, Monday, boy, Tuesday, child2)
(boy, Tuesday, boy, Tuesday, child2)
(boy, Wednesday, boy, Tuesday, child2) ...
(girl, Saturday, boy, Tuesday, child2)
(girl, Sunday, boy, Tuesday, child2)
So we do have 28 possibilities, because the case of two boys born on Tuesday is in fact two distinct cases, depending on whose child the question refers to.
That what would be for example that outcome of randomly selecting parents of two children, making them select randomly one of the two kids, and ask the question considering that children.
Which also has the positive effect of making the maths match with common sense (that's not always the case, but is always pleasant) : why would a totally unrelated elements like Tuesday change anything in the probability ? With no precision of weekday it would be 1/3, with weekday 13/14, with other unrelated stuff it would keep changing the probability ?