We are a family of four and our 2 boys were about 9 and 13 when we did this cruise.
For Reykjavik: the Golden Circle tour! ours included also a tour of a geothermal station, so it lasted about 12 hours total if my memory serves me correctly, but was totally worth it! The National park and Gulfoss waterfall were spectacular, and the Geyser area was fun (with nice shopping opportunities)! Also, the Blue Lagoon! sure, it's touristy, but both the adults and kids got a total blast out of it! For what it's worth, we liked Iceland so much that we went back by ourselves the following year and went to the Laugarvatn spa. It is also beautiful and is more authentic of course, by a nice lake, but Blue Lagoon is much, much bigger and much more spectacular.
For Akureyri: we went horseback riding by ourselves, at a place with absolutely gorgeous scenery, called Pólar Hestar:
www.polarhestar.is/
They picked us up at the port, took us for a short visit to the very interesting Laufás historic farm nearby , and then we had an hour of horseback riding (they had longer tours, but our youngest was about 9 yrs old so we didn't want to overdo it; this was also the reason we booked directly through them instead of with DCL, because DCL wouldn't allow children younger than 10 to book a horseback riding excursion)... Oops, I realize you are not going there, but will leave the comments there in case they are useful to someone.
For Stavanger, we did the Iron age farm: it was ok, the children liked trying archery and running around, but I wouldn't say it's a must, whereas my parents did the Lysefjord cruise and raved about it.
For Kristiansand, we went walking in the town and did the open air museum, which was nice, especially for the kids. The town is really nice, we had a bite to eat in the harbour at a nice restaurant called Bølgen & Moi.