EDIT: I just now realized that your comment above is a great example of the sort of follow-up questions I'm talking about. Well played.

Examples from the past week:

• Started with the "how was your week" thing. The guy had been on an MIT board discussing their strategy for building MOOCs, and I got to hear a lot about business models in education and how that's changing with technology.
• Him: "I'll be leaving early tomorrow." Me: "Where are you going to be?" Well, he's helping his son move, and also trying to deal with the previous landlord because apparently his grandkids damaged the walls, and there's all sorts of drama around that...
• I overheard someone talking about hockey. I know absolutely nothing about the sport, so I asked some extremely basic question, I can't recall what. I learned a little about the structure of the game, and then a lot about how stricter enforcement of the rules in recent decades has changed the dominant playstyles.
• Right now, in my IRC window, I am hearing about changes to World of Warcraft in the ~5 years since I've played after asking about a cryptic comment someone made about downloading a patch.

As you can see, this is at best an imperfect tool for getting a specific type of story. The core of the technique is that I don't have anything in mind when I start, and I'm not steering towards any particular topic.

I haven't tried to get stories like the ones you're looking for, but I've found that being direct is usually a good approach. Maybe just go with "so who are the important people in your life?"