This claim is based on a paper by Tatiana Zerjal titled The Genetic Legacy of the Mongols, which appeared in the American Journal of Human Genetics.

Couple of things to clarify first:

• Genetics can't narrow down descent to one person who lived so long ago. What the paper actually says is that about 1/200 men alive across the world today have an ancestor who lived in Mongolia about a thousand years ago. We can deduce that this person must have been politically powerful and prolific, so Genghis is a good candidate.

• This marker was found on the Y-chromosome, so we're talking about patrilineal descent only. We haven't actually discovered Genghis Khan's grave so we don't have a sample of his DNA. The originator of this lineage might have been Genghis Khan's father or grandfather or great grandfather, since they fit the time frame as well as Genghis himself.

So the genetics evidence says that some Mongol male who lived about a thousand years ago in Mongolia left behind a lineage that has grown quite tremendously in the years since, and now includes about 1/200th of the world's population. How did this happen?

In order for a lineage to dominate others, it must consistently have more individuals who survive to reproduce than other competing lineages. So the short answer is that more males in this lineage survived to have children than was the world average at those time points.

Note that we're not saying that it's because Genghis or his sons had a lot of children (though they did). That alone wouldn't account for it. We're saying that for ~35 generations since Genghis, his descendants consistently had more children who survived to reproduce than the world average. The 1 in 200 figure is the result of selection exerted over 1000 years, not due to the large number of any one person, whether Genghis or Kublai or someone else.

How much selection pressure is needed to account for these results? If you average it out over 34 generations, then each generation produced 36% more male offspring who reproduced than other people around them. That's a very high number which can't be explained by chance or genetic drift or any biological factor, so it must have been social factors which caused this lineage to be so prolific.

This is where history mixes with biology. The Mongols created a huge impression on people all over the world, an impression that was powerful and long-lasting enough that kings and rulers gained legitimacy for centuries by having a connection to Genghis Khan. Being able to trace your ancestry to Genghis gave you legitimacy and put you at an advantage, whether you were a king or tribal leader or just war chief. This advantage lasted for hundreds of years after Genghis died.

So the reason why Genghis Khan's lineage did extraordinarily well is because for hundreds of years, his male descendants were powerful kings and leaders across much of the world. To some extent, this may have enabled them to have more children through more wives, but the real reason for the difference is that more of their sons survived to reproductive age. Infant mortality was terrible in the old days, so even a small advantage that a child might gain from a wealthy and powerful father could produce a significant difference in survival rates.

If each of the generations since Genghis produced 36% more sons who survived to have children of their own, you can explain the 1 in 200 Mongol ancestry attributable to him.