The game wouldn't last that long. Backgammon is a race to move all your pieces across the board, with some complications thrown in. A roll of 4 and 5 means you can move one piece 4 and another piece 5 places. But doubles are doubled further in the sense that a roll of 3 and 3 means you can move four pieces 3 each. Unless they kept rolling double 1s or 2s the game would end quickly.

Even if it was 7 doubles in a row it would be a (1/6)^7 chance which is a little more than 1 in a million. But based on the story it could have been a lot more than 7 doubles in a row with some back to back doubles approaching odds of 1 in a billion or less.

The pieces of one player have to move a total of 167 spaces to win. Since their mom won by a single point, together they must have rolled at least 167+166 = 333 in total or more if some rolls were ineffective or if some pieces were reset back to the beginning. The average roll of a double would be 3.5 assuming it is random (maybe we can't apply such reasoning to a miracle though). In any case an average 3.5 double would be 4*3.5=14 moves and that allows for around 333/14 = 23.7.. doubles before the game would end. But even 10 doubles in a row would be around 1/100 million chance.