The Lucky Quarter is an item carried by Travis Grady in Silent Hill: Origins. Having acquired it before his journey into Silent Hill, it seems to serve no physical purpose, though it does prove to be the first piece of information needed to solve the Calendar Puzzle, with two of the numbers needed written on the coin.
The Lucky Quarter is the same coin that Travis kept from playing games in Riverside Motel, and couldn't return it to his father when he committed suicide. It is likely that the '61' etched into the coin refers to his father's year of death, 1961. This would signify that Richard did not commit suicide on impulse, and that this coin would mark a permanent symbolism of his memory.