The word "backgammon" was not invented until the 17th century. Both Shakespeare and Chaucer referred to the game as "Tables" in their works. Tables is a derivation from its longer Roman name "Duodecum Scripta et Tabula".
Shakespeare died in 1616 and "backgammon" first appeared in print in 1645 so somewhere between those two dates "backgammon" came into being but nobody knows where from.
It is also generally acknowledged, but not proven, that when the game changed from Tables to Backgammon two things happened. The use of three dice to play the game was changed to just two dice but when you rolled a double you were allowed the play them twice, e.g. if you rolled double twos then you could play four twos. This compensated for the loss of the third die from Tables.
Pictures of tables/backgammon from the 16th and 17th centuries are rare but I have posted two of them here to support the fact that three dice were still in common use at that time. The first is a miniature that I spotted this week on Getty Images. It is taken from the 1283 picture 'Libro De Los Juegos' (The Book of Games) by Alfonso X, called The Wise (1221-1284), Three dice are clearly shown and very rarely, both players are female.
The second one is much more famous as it is a detail from Hieronymus Bosch's "Garden of Earthly Delights (Hell)" which was painted between 1490 and 1510. Again the three dice are clearly visible.
The key questions, which sadly I assume will never be answered, are:
1) Who created the word "backgammon"?
2) Who changed the game from using three dice to two and introduced the idea of playing doubles twice?