Calculating luck as a factor in board games?
A reasonable empirical measure of luck is the probability that the best player can beat a good player in a given game.  So if you can get data about a large player population (I have done this with Race for the Galaxy), you can get an implicit measure by looking at the Elo rating difference at various skill percentiles.

On the other hand, this only tells you about the amount of luck in the game as a whole, not in a particular game.