Twilight Struggle
"Twilight Struggle" was the highest-ranked game on BoardGameGeek from December 2010 to January 2016. As of August 2018, the board game occupies a fifth spot. Being released in 2005 it is the oldest board game in the overall top 10 ranking.
Nine men's morris
According to R. C. Bell, the earliest known board for the game includes diagonal lines and was "cut into the roofing slabs of the temple at Kurna in Egypt"; he estimated a date for them of  BCE. However, Friedrich Berger wrote that some of the diagrams at Kurna include Coptic crosses, making it "doubtful" that the diagrams date to 1400 BCE. Berger concluded, "certainly they cannot be dated."