I'm from another part of Norway, but lived in or around Oslo for 10+ years. I'm not sure if you're familiar with "Nordmarka"? It's a somewhat protected area, with a lot of great hiking spots. I don't know if you drive or not, but if you don't, you could still get there easily by train from Stockholm, then public transportation from downtown Oslo.
Since the area is pretty large, you have a lot of options for how far to hike before setting up camp, if you want to remain close to trails or roads etc. My kid got used to hiking and sleeping in a tent up there, and we have lots of good memories from around there. Drop me a message if you want in on our favourite spot.
If you're a bit more adventurous and the kids are a bit older, western Norway has some good places with fjord-views.
If you want more mountain, you could look at Hardangervidda, between Bergen and Oslo. Finse for example, is reachable by train, and from there you can hike to other cities also along the train route, so you wouldn't have to make it a round trip.
Those should give you some good options of various degrees of difficulty, but in all honesty, you're likely to find something closer to home as well, if you'd prefer that.
If you'd like to look more at places and routes in Norway, check out ut.no. It's a pretty good selection of hiking trips/routes, well integrated with pretty good maps as well.
Normally I'd remind you to bring enough lights for everyone, but we are on CPF. ;-)