It would very difficult to come up with a metric that depends solely on the game itself, and not the population of players. It's much easier to answer the question "How large a role does luck play, in comparison to the skill difference of players?" than to answer the question "What is an absolute measure of how much of a role does luck play?"
If it's defined in terms of the players, then there are a variety of metrics. One would be to look at the variance of win rates among players; the higher the variance, the less important luck is. However, you'd have to adjust for the fact that players tend to play players of similar skill, which decreases the variance; if two players of exactly equal skill play, then the game will come down solely to luck. We could also look at how much Elo spread there is (different games use different parameters for their Elo assignments, but if you construct Elo scores using the same parameters across games, then comparing the Elo spreads should give some sense of the size of luck).