There were reports in the news recently about findings by researchers at the University of Copenhagen that show that all people with blue eyes share a single ancestor. This ancestor was supposedly the first person to express a gene mutation responsible for producing blue eyes; he or she lived in eastern Europe (probably around what's Ukraine today) between 6 and 10 thousand years ago and all persons living today who have pure blue eyes (as opposed to blue eyes with brown spots/rings) are related through him/her.
Well, I have blue eyes (so do Jennifer Aniston and Cameron Diaz; hi there, cousins!), and I mentioned this report to my wife; she asked "how do they know that?". I started to answer something about looking at genetic details and then I realised... I didn't really know. So I went after this information.
Looking for this on the net I found many copies of the same press release, with small changes here and there. It took me a while to find a link to the original article with the research results, and I have to say that it's not an easy read for someone (like me) who is not a researcher in the area. Still, a few things were somewhat clear, and merging information from the article and from the press release I think I understand how they came to their conclusions.
So, the answer to "how do they know?" is, they actually don't: they assume this is what happened, with a very high degree of confidence, due to what they found in the genes of the people they've surveyed.
Basically, the "normal" eye colour for human beings is brown; up to the moment when the blue-eye mutation first occurred, everybody had brown eyes. There's a large degree of variation in the gene that codes for brown eyes, which indicates that it's an "old" gene (in fact, it is shared with many other mammal species with brown/yellow eyes — other primates, cows, cats etc.). This also has the effect of introducing large variations in the actual colour of the eye (from pure brown to very dark brown almost black, to hazel, to blue with brown spots. to blue with brown ring around the pupil, to grey, to green — yes, green eyes are a variation of brown), due to differences in the amount of melanin in the iris.
People with blue eyes, on the other hand, display almost no variation in the gene coding for eye colour (and, hence, in the amount of melanin in the iris); they (well, we) all display the exact same mutation in the same location of the same gene, which causes an iris almost devoid of melanin. This indicates that the mutation is very recent and, at the same time, point to what the authors refer to as a "common founder" mutation - that is, a mutation that spread from a single person. As far as I could tell from the article, the date and geographical location of the original mutation is determined from the prevalence of the mutation in current populations and our knowledge of human migrations in the last 10,000 years.
Interestingly, the article does mention that the very high frequency of blue-eyed individuals in some areas of the globe (say, Scandinavia) indicates that this is a trait that is positively selected; that is, blue-eyed individuals have historically had a better reproductive success, at least in these areas (and this makes it harder to calculate the age of the original mutation). The reason for that advantage is not clear; it could be related to vitamin D absorption in high latitudes, or even to sexual selection (blue-eyed people being more successful in attracting partners). Your guess is as good as mine (and the authors').
So, here it is; that's how they know that.