Hi! First, I like your approach: Don’t try to see it all!
The South Coast definitely would be perfect for you, except that it’s pretty touristy. Still, I think it’s okay to go for it in the summer. You can choose to hit some of the outdoor attractions early or late, when the crowds are lighter. (Use the 24 hours of light to your advantage!)
If you do choose the South coast, book lodging now! South Coast lodging fills up very quickly.
Another option is to head to the Snæfellsnes peninsula. Here are some of our live updates from there (the first one written by my son!):
https://icelandwithkids.com/2016/06/22/sunday-june-19-update/
https://icelandwithkids.com/2016/06/22/monday-june-20-update/
All of that is on the southern part of Snæfellsnes (besides the museum at the end of the second post), and are less than an hour of driving from start to finish. Well, once you get up there.
You could head around to the north, and get some glaciers in on the way:
http://sfn.is/tours/glacier-lava/
(I don’t know anything about that tour, other than it looks great and the ages work for your kids!)
Up north there are some waterfalls too.
You could do Reykjavik, the Golden Circle, and then a few days in Snæfellsnes; I think that would make for a great trip without much driving.
So maybe check out our South Coast of Iceland touring plan and see what you think. Then decide between more time there or heading up to Snæfellsnes.
Either will be great. Good luck, and hope that helps!