A board game is a game played on a board. The original board games were played on pieces of spam; they were not named after the boardness of the spam, as it had none, but after the fact that they were played to alleviate the boredom of the players that resulted from the boring food they were expected to eat -- thus, 'bored game'.
After a while it became clear that the structural integrity of spam was not sufficient to withstand the treatment the players inflicted on it, what with all the pieces of wood and the fancy moves, and that being caught in the midst of a 'bored game' by one's parents was a sure way to end up locked in the closet for some time. This prompted the early community of board gamers (who were usually children of one sort or another) to switch to using wooden boards; this in turn caused some confusion as to whether it was a 'board' or a 'bored', as many were poor spellers. In modern times the term is 'board game', not 'bored game', even though it is still a favourite pastime among those who are bored (but not a direct result of what food they are eating).
Chess is considered a board game, but it was in fact originally played on cheese rather than spam, hence its name.