Fix the Budget - There is one obvious area where you could make a statement with a game that you can't make (the same way) in any other medium: Manipulation of the game mechanics. Sim City, in a way, taught me the importance of balancing a budget better than any book or article. In fact, the Sacramento Bee had a somewhat simpler "game" a couple of years during our budget crisis where they simply list programs you can cut funds from and programs you can add funds to, and you have to cut a total of $1 Billion from the budget: Fix the budget mess - your way.
The idea was to make it clear how important each program was, and how hard it was to cut funding from them. So whenever you cry out about some vital program's budget being cut, you have to think: Well, if they restore funding there, which other vital project's funding should be cut instead? My point is that the game did not need to explicitly spell out this moral, like I did. The mechanics of the game itself spoke the message.
Conquer the World - In a game like Civilization, your goal is to conquer the world. In the original games, you need to balance civic and scientific pursuits with military ones, but, ultimately, your science serves your military. You need territory to win, and you get territory by taking over your neighbors. It's interesting to think about what statement this is making. In Civilization III, they add "culture" as a way of expanding your civilization and taking over neighboring cities non-violently by converting their minds. That's an interesting twist, but, ultimately, the goal of the game is still to conquer the world. By making that the goal, the game subconsciously makes you relate to that goal, and to think of it as a "good thing" to some degree.
People apply this idea to video game violence, and while I don't think video game violence is solely responsible for increased aggression and whatnot, I can see the basic statement that many video games make: Violence solves problems. I think they make that statement partly in jest, though, and partly for fantasy-fulfillment that most people can differentiate from reality, so of course it's complicated... But I do think there's something to be said about games that encourage you to think your way through problems.
Wild West Bank - There's also this game called Wild West Bank that I found a while ago. To play it, click the sign with the red letters to start, then click on any trailer or double-click on any house to get rid of it. However, if there are any soldiers guarding a house, you have to first drag them out of the West Bank (which seems to make them happy).
Eventually, the settlements start popping up at a faster and faster rate, such that you can't dream of keeping up with removing them, and so you always lose. Again, the game mechanics speak the message in a way that words can't. The game was created by an anti-settlement group.
Also, both of these games do something else that words can't: They put the player directly in a role. The games define a goal for the player, which causes the player to identify with that role, because they're now an active participant. When you fail to balance the budget, you might think, "This is tough!" and relate to our state legislators a bit more. When you fail to stop the settlements, you might think, "Oh no! This is impossible!" and relate to the anti-settlement people more.
Photopia - This reminds me of Photopia, which uses the basic mechanism of those old text adventure games, but in a novel way. Try this web-based version. (Just note that you have to type into the text box below, and to use [enter] when it says "hit space", because it was originally designed to be played on a PC.) It's in the format of a typical text adventure, where you type commands like, "look", "look at treasure chest", "north", "get shovel", "talk to salesman", etc., but it uses the medium in a very novel way. There are very few puzzles to solve, and it's quite linear, but it really focuses your attention on what interactivity can bring to the table. I urge you to try it out.
Photopia illustrates my last point especially well: When you actually play a character, it makes you relate to that character and their actions more so even than when you read about them in the first person. In fact, I was going to post some spoiler comments here, but I'm going to do that some other time instead, to further encourage you to try it out for yourself. I'll probably take you less than an hour to get through it if you're familiar with text adventures, and maybe twice as long if you're not, but it's totally worth it. It's probably the most innovative and moving piece of interactive fiction I've seen.
I'll give you guys a week to try it out, and then I'll summarize and talk about it. But really I urge you to try it for yourself first. :)