I am interested in developing social intelligence in my child. What behaviors have parents introduced into their daily family routine specifically to encourage the development of social intelligence?
I think a daily family supper is a good idea. It's an opportunity for everyone to talk about their day, and how they feel that evening.
Listening, and empathizing, as well as thinking about it and commenting should be a start towards what you're looking for.
I think the best thing to do is get your child thinking and talking about social situations. Presenting your child with hypothetical stories, role-playing exercises, and discussing real stories of people you know and people in the news are a great way to introduce social situations to a child (or adult).
In particular, discussions about ethical or moral issues with hypothetical characters or real people are a great way to get kids thinking. I would try to be as Socratic as possible when doing this. If you just tell them the "right answer" from your own perspective you might be teaching them to blindly copy what you think, or perhaps worse yet, take the complete opposite view. If you focus on asking good leading questions you can really get them thinking about the issue at hand.
Examples of situations, people, or hypotheticals that you can discuss:
3-5 years old
5-8 years old
8-18 years old
We try to spend our time at dinner or during long car rides discussing different issues or hypotheticals. I find it is best not to discuss people who are actually part of the discussion. This often leads to unanticipated hard feelings, defensiveness, or teasing.
Our children seem to respond fine to this type of discussion, but some kids don't. One friend came up with a creative way to having these discussions with his kids. He would play role-playing games (old-fashion RPGs like D&D, not video games) with his sons and introduce different social situations to which his children's characters had to respond. The RPGs also taught his kids to collaborate on a shared goal rather than competing against each other (as @torbengb described).
I've tried to work on this with emotional intelligence: helping my kids understand what they and other people are feeling. Other than that, I try to have them interact with a wide range of different age groups and participate in group activities like sports.
Looking for activities to do in the family, and of course depending on the child's age, you might want to consider board games or garden games. Many games are competitive (players play against each other) but some games are cooperative instead, just like (the own team of) a ball game would be.
I wish I could mention some specific suggestions but I can't think of any right now.
First, Social Development is a pretty broad topic and a lot of it happens fairly naturally. Even gender identity falls under this category and we all develop one at some point. At the same time, you are right to want to specifically address this with your kids. You should know that one of my basic assumptions is that good manners, good social graces and social awareness are all closely linked. A person who is empathetic, thoughtful and fairly self-aware, will automatically exhibit good social graces. With that in mind, I think there are Three major things you can do with your child at almost any age to help with Social Development in significant ways and one more you can add once they reach age 7 or 8 or so.
The "Seven Habits Series": Of interest to families, "The Seven Habits of Highly Successful Families", "The Seven Habits of Successful Teens" and "The Seven Habits of Happy Kids" (that last one is FABULOUS for elementary kids).
10 minute Life Lessons For Kids by Jamie Miller: This book includes fun activities and "games" to play to get a better grasp of concepts relating to Honesty, Trrust, Love, Empathy etc. - Great for Primary School.
E is for Ethics by Ian James Corlett: This book gives 26 quick stories with discussion questions to help get a conversation going after the story has been read. Great for Preschool through about 2nd or 3rd grade.
Aesop's Fables: The version we have was illustrated by Nora Fry: This is a wonderful collection of classic stories that illustrate just about any type of quality of character. You can read a story and after they are fairly familiar with what a fable actually is, you can even have them predict what the fable will be before you read it to them. Sometimes its interesting what they get out of a story vs. what you do and what Aesop intended. Then you can talk about modern examples of stories that would show the same lesson. Besides, these stories are just good literature anyway. If you are religious you can relate these discussions to your core values within your religion as well.
While watching movies or reading books where a child (or adult) is behaving in a not-so-gracious kind of way, pause the movie and ask your kid questions like, "what do you think about that choice that child just made?" "What do you predict will happen to this child later in this movie because of this choice?"
There are TONS of great movies and picture books out there about all kinds of social graces as well as issues. For example, Charlie and the Chocolate Factory gives some great examples of some pretty anti social behavior (social graces), while Fly Away Home is a great picture book by Eve Bunting that shows a child and father that are homeless and live at the airport (understanding social issues). Then there is also a host of wonderful books about manners out there as well. I especially like "Do Unto Otters" by Laurie Keller for little ones.
And the last one, that has a specific age: When they are old enough to become civically involved, get your children involved. Civic duty is as much a part of social development as many other things. That can mean (depending on ages) helping with a beach clean up, creating an education campaign about a pet issue of yours to going with you to the polls when you vote and learning about that process and how you think through the decisions you make when you vote.
Have fun, and let us know what else you find please.