Clannad: Have the player make choices during the game that affect sub-stories. Simple things like which person to ask for help or whom to pay a visit sometime or whose advice to listen to or whatever. Allow us to explore the backgrounds of minor characters if we decide to make choices that focus on that particular character. It would add a layer of depth and give the game replay value.
Maybe even make it so that you can choose to help them with their personal problems, and if you make the right choices, that character has a "happy ending" for their own story, while if you make the wrong choices, things remain unresolved. Just don't go overboard with rewards for these kinds of quests - they're supposed to be nice touches, not game-changers.