When I posted this problem on my blog, one commenter (who prefers to remain anonymous but gave me permission to repost here) noted a cool way to estimate the expected value of $B/(B+G)$.
Write $f(G)=B/(B+G)$ and expand in a Taylor series around $B$:
$$f(G)=f(B)+f'(B)(G-B)+(1/2)f''(B)(G-B)^2+...$$
Now take expected values: We have $E(G-B)=0$ and $E(G-B)^2=2B$, so
$$E[f(G)]=f(B)+(1/2)f''(B)(2B)+...$$ $$=(1/2)+1/(4B)+...$$
Now the number of boys is equal to the number of families, so for $k$ families, the proportion of boys is well estimated by
$$(1/2)+(1/4k)$$
and of course it's easy to get better estimates by going to higher terms in the Taylor series.
My commenter also adds the following (in my opinion, quite insightful) remarks:
Independence (or, more precisely, correlation) isn’t the only issue. Even for independent variables, the expected value of a ratio is not equal to the ratio of the expected values. (The expected value of a product of uncorrelated variables is the product of the expected values, though.) This is one of the most important keys to understanding this problem, I believe. And this is why I suggested the Taylor series to expand the ratio about its mean. I also think it is a little easier to find the expected proportion of boys because the random part (G) only appears in the denominator. Also, B is equal to the number of mothers, so I don’t believe B and G are independent because I don’t believe the number of girls is independent of the number of mothers.