A Game of Thrones is one of those titles that appears in the book over and over… and over. After all, "When you play the game of thrones, you win or you die. There is no middle ground." (46 Eddard 12.80). Yikes.
Even if it seems insignificant, it's worth noting that this is "a" game, not "the" game. No matter who wins this round, there's always another game of thrones to play. Which makes sense: the Targaryens won and became kings… until they lost and someone else took over. And so it goes.
As for this being a "game," it's well worth keeping in mind that, in this game, people die. It's not pretend, or just for fun. In fact, people who aren't even playing can die just as easily. Just think of Masha Heddle. Ouch.
Let's not forget that this is part of a series called A Song of Ice and Fire. We're pretty sure this name seems to promise some elemental clash. Maybe the Others are Ice (they are cold and live in the north) and the dragons are fire (because they, you know, breathe fire). And they're going to form a band and go singing. No, wait, they don't form a band. Martin calls it "a song" because a lot of old myths are told in the form of songs. (You totally win cool points for knowing that.)