What does meaning really mean? Generally it translates as resonant, illuminating, symbolic or significant. In some cases all of the above. A meaningful song might evoke the history of a revolution for the listener, so that even though she does not know the facts she feels a connection to it. The same is true of novels, movies and art.
Games incorporate agency and so many of the events that happen within them are of a player's making. An action causes change in the game world, and can therefore be significant, but not necessarily resonant, symbolic or illuminating. The question for games is really whether they can incorporate other kinds of meaning too.
I think they can.
Meaning is sometimes shared, but subjective and hard to explicitly define. A nation's flag is often considered meaningful, but the meaning that it holds for one group may be very different to another. Minecraft art can be personally very meaningful for the community that created it, or it can end up being a case of 'death, terrible as prunes'.
While we often think of meaning and tearful emotions as one and the same, meaning is actually much broader than that. The weight of symbolism attached to the election of Barack Obama made many people cry, but not all meaningful experiences do this. The Mona Lisa never made me cry, nor Joyce's Ulysses or Shakespeare's Merchant of Venice. Damien Hirst's Physical Impossibility of Death in the Mind of Someone Living is not the sort of art work that brings forth gut-wrenching sobs, however it does make you think.
Making people think is, I suspect, the key to games as art. Games are usually better at illumination than emotion. It is hard to push emotion at players because of the tendency of their play brains to reduce games to objects and levers, but games can be instructive and can play on symbols (particularly signifiers) for the purposes of commentary.
One example of a meaningful game for me was Credo, a little-known title from Chaosium. The game was about the formulation of the official doctrine of Catholicism in the council of Nicea. It involved a lot of horse-trading between players over the roots of catechism, resulting in some entertaining versions of what would then become the New Testament. For me it was meaningful as someone who grew up in Catholic Ireland with all of the foibles that that entailed. It satirically illuminated theology in a way that I had never considered, and the game would make me reflect on the things to which society gets attached.
Many other games have done likewise, using the inner workings of things and acting as a kind of commentary. The PC strategy game Fate of the World sets up a situation of resources and pollution in crisis, based on copious scientific data. But you draw your own conclusions. I even find something quite meaningful about the progress from bears to churches to temples and finally a treasure chest of money in Triple Town. It sort of says something, when I think about it.
As a part of exploring the world and believing in its numina, games communicate ideas and capture moods, even accidentally. When Insomniac used Manchester Cathedral as the setting for one of its Resistance: Fall of Man levels, it was probably not their intent to make a commentary. However it upset some people anyway, tapping into ideas they held as meaningful. Traveling through the sands of Journey and communicating with other players can be meaningful, as can the sense of story throughout. It is illuminating and symbolic, but never explained. Realising the extent of the Sims system and the dark purposes that it can be used for is certainly satire, but often quite meaningful too.
So there's endless scope for making meaningful experiences happen with games, in a sense allowing it to be discovered or self-created. As game makers we're not necessarily authors, but sometimes the enablers of authorship. That places us in a unique position.
(Today's image is of the game Journey from thatgamecompany, a game which I found very meaningful)