The reason it's not "easily seen" in my way is that this is incorrect.

Under normal circumstances, if the woman has two children there are four possible outcomes:
b-b
b-g
g-b
g-g
Each outcome is equally likely, that is 1:4.

Damocles' challenge says that we know one of the children is a boy, so we eliminate the fourth outcome:
b-b
b-g
g-b
Each with a 1:3 likelihood.

Exactly one of those outcomes is that the woman has two boys. Accordingly, the probability of the woman having two boys amongst two children given that she has at least one boy is 1:3.

Jibb

EDIT:
@Quadraxas: Actually, the challenge states that at least one of the woman's children is a boy.