She makes the same claim in her book, Reality is Broken: Why Games Make Us Better and How They Can Change the World. There, she gives a reference:
Rawlison, George trans., with Henry Rawlison and J. G Wilkinson. The History of Herodotus: A New English Version (New York: D. Appleton, 1861), 182 http://www.archive.org/stream/historyofherodot01herouoft#page/180/mode/2up
It can be confirmed by following the link provided that, assuming this is not a fake that somehow got into archive.org, the person writing under the name Herodotus indicated that the Lydians believed themselves to be the true creators of all games they share with the Greeks:
The Lydians have very nearly the same customs as the Greeks, with the exception that these last do not bring up their girls in the same way. So far as we have any knowledge, they were the first nation to introduce the use of gold and silver coin, and the first who sold goods by retail. They claim also the invention of all the games which are common to them with the Greeks. These they declare that they invented about the time when they colonised Tyrrhenia, an event of which they gave the following account. In the days of Atys the son of Manes, there was great scarcity through the whole land of Lydia. For some time the Lydians bore the affliction patiently, but finding that it did not pass away, they set to work to devise remedies for the evil. Various expedients were discovered by various persons ; dice, and huckle-bones, and ball, and all such games were invented, except tables, the invention of which they do not claim as theirs. The plan adopted against the famine was to engage in games one day so entirely as not to feel any craving for food, and the next day to eat and abstain from games. In this way they passed eighteen years.
(pages 181-183, heading 94).
So IF Herodotus is accurate in his account, and IF the Lydians are accurate in their telling of the story, then they invented dice, ball, and bone games, though not all games. The general shape of the story matches her account, but it's been embellished somewhat to make a point.
However, in the footnotes of the translation McGonical cited, the following note can be found:
The ball was a very old game, and it was doubtless invented in Egypt, as Plato says. It is mentioned by Homer (Od. viii. 372), and it was known in Egypt long before his time, in the twelfth dynasty, or about 2000 B.C, as were the [...] counters, used in a game resembling our draughts, with two sets of men, or "dogs", of different colours.
Therefore, while it is true that Herodotus said this, it is doubtful that he was correct. Logically, it follows that if they didn't even invent ball and dice games, they did not invent all games.