As proof of the masterfulness of this troll, he now has about 750 comments on this post, which is about a factor of five higher than his usual. I can only assume this has been reddited and slashdotted and discussed in comments there as well. People have all sorts of answers, and they are by and large completely convinced of the irrefutable accuracy of their answer.
The actual numerical answer, of course, is 50% -- on the grounds that there are only a few meaningful probabilities in the real world. These are the once like 90% which means "pretty sure", 10% for "I doubt it", and in this case 50% which means "I have no clue one way or the other." (It does not, however, mean that this can be treated as equivalent to a coin flip for purposes of fair gambling.)
Jeff even alludes to this answer in his title of the post, calling it "The Problem of the Unfinished Game".
Consider that, in the information you're given, there is no mention of how you met this person.
If you assume a population of parents of two children of gender selected by probabilistic equivalent of a coin-flip (which nearly all of the respondents do, even though the actual distribution is not 50-50 and one ought to at least defend the assumption of non-correlation), and select all of the parents for whom one of the children is a girl -- let's say you meet parents on the street, and talk to each of them until you find one who meets this criterion -- you get an answer of 2/3.
If you assume the same population, and select a random girl and talk to her parent -- let's say you send your daughter to an all-girls age-segregated day-care, and talk to parents of other girls in the same class -- you get an answer of 1/2.
If you assume a population of parents of opposite-gendered kids -- and why shouldn't you? Maybe you're at a support group for them? -- then you get an answer of 100%.
Or, if you assume the real world, in which conversations that contain this particular bit of information but no information about the gender of the other child happen somewhat rarely, and their occurrence can be expected to have all sorts of correlations with the gender of the other child, and moreover you are selecting these people through some probably-not-entirely-random selection mechanism which Jeff has told you utterly nothing about -- who knows what you'll get?
(I think Jeff currently only has one kid, or I'd say the answer is "100% if Jeff has a daughter and a son, or 0% if he has two daughters.")
I somewhat despair of the implications of what it means that many people think that math alone can give them the answer to this question. Sigh. Grumble. Et cetera.