It would be nice to see a game with the protagonist already in an established relationship with someone, a relationship that serves to add depth to the character but is not immediately used either as a plot tool or as the functioning premise of the game. Catherine (anime trope-isms and all-around lack of understanding of the way humans work aside) is a step in the right direction, but it's not what I'm getting at based on what I just described. Why not have a game where player control shifts between the player and the significant other depending on the point in the game you are at? And I don't mean as something level-based. It could even be manual, where the characters have different strengths that can either be instructed (though less effective considering it's AI) when you're not playing them or controlled directly when you are, where you jump between them and therefore have more control not over their gameplay but their interactions and responses? Why not have a game where the missus (or the mister for the politically correct) even as an NPC is not always off-screen as a goal or something to be obtained and instead serves as a constant interaction for a protagonist who does not have to be a silent husk--you know, a 3-dimensional character? In that regard, their relationship isn't what gets the game going, but keeps the characters accessible and compelling. Of course, that requires good storytelling and an understanding of all the possible ways a player might choose to interact with that person. Sounds like hard work, right? But are we really fine with not asking games to work harder to keep us interested 30-something years after gaming really took off?
You know what would be an awesome game? Think of a family like that in the Incredibles (no, I don't mean a game adaptation of the movie, just a concept of an entire family united for a common goal), and set it up where they all have individual pros and cons but all can be controlled by the player; I don't even necessarily mean that you need to give them powers, just things they excel at and whatnot. Make some parts of the game where you only have access to one, but most of the game where you have access to all of them. And make it so that there's no one correct way to go through the obstacles presented in the game. And make some choices easier or harder based on who you interact with and how. You know, it'd be like the Incredibles meets Sam and Max meets (at least from what I've heard about the game) Deus Ex.
The depth and complexity it would require to make such a game come to fruition is immense, I'm sure. But I cannot honestly think of such a concept not being a damn great, timeless game. Or at least the stepping stone for whatever game truly achieves that status.