Splitting 8 nights between Copenhagen and Oslo

It depends. What are your interests?
Are you going to take your kid to Legoland and/or the Lego House in Billund? If so, you would need to figure out how you are getting there and back and whether you need to stay overnight. There is a little airport there, so maybe you could fly from Billund to Oslo (I didn't check on the availability of this) so you don't have to backtrack to Copenhagen.
The best way to determine how much time is needed in a place is to make a list of things you want to see and do and figure out how much time is needed for each, then do the math. A very fun thing for a 7yoa child is hunting for Thomas Dambo's trolls. https://www.visitdenmark.com/denmark/things-do/family-holidays/where-find-trolls There are lots of blog posts, etc., giving detailed information about how to do this by bike and public transport.
You also don't say when you're going. Next summer, when the days will be long, or next month, when the days will be much shorter but not as short as they will be in, say, December?

If you have 4 full days for Copenhagen I think adding another day to Oslo might make sense. Some things you and your 7 year old might really like in Oslo:
1. Frogner park, even if your child isn't interested in the Vigeland sculptures, the park is beautiful with plenty of room to roam and run
2. Ferry to Fram. You take a little ferry boat to the Fram museum. Once there, the Fram is super interesting because you can climb aboard and go through the whole ship
3. Ferry to little islands
4. Roaming around the Aeskershus fortress

In Copenhagen definitely take a canal boat ride, definitely allocate plenty of time for Tivoli gardens. Enjoy!

I took my 10-year-old son to Copenhagen and Oslo, spending 5 nights in Copenhagen, one on the ferry, and 3 in Oslo. Caveat that this was back in 2013, but I think the essentials haven't changed much. Certainly another night in Oslo would be worthwhile, but there is plenty to occupy you in Copenhagen for 5 and your two full days in Oslo will allow you to see quite a lot there. You wouldn't really go wrong either way. Here are the Oslo things we enjoyed: Akershus fortress including Resistance museum; ferry to Fram Museum (there is also another maritime museum there and the Kon-Tiki raft; unfortunately the Viking ships museum is currently closed for restoration); Frogner Park; a look inside Oslo town hall; Munch paintings at art museums (even if your child is a bit young for full enjoyment of art museums, he or she might like these); and a trip on the T subway line to the station at the end of one of the lines for a nice relatively short hike around Sognsvaan Lake. In Copenhagen: canal/harbor cruise; Viking ships in Roskilde; Kronborg castle in Helsingor (very scenic and historical); Tivoli; the National Museum especially Viking exhibits; Copenhagen zoo; an excellent aquarium (de Bla Planet); the big Lego store in central Copenhagen (I did not consider the very lengthy side trip to Legoland in Billund). The overnight ferry was a great way to get between these cities, and the way I looked at it, it basically was free transportation, because a cabin plus a nice buffet dinner (reserve early seating) was about the same price as a hotel room and dinner would have been for that night in either city, perhaps even less.
I found a multiday Copenhagen Card to be an excellent bargain: Free public transportation in the region, included admission to virtually everything we wanted to do including some otherwise expensive things such as the zoo, aquarium, and Tivoli (park admission but not the rides), and a child card was free with purchase of an adult one. However, you would want to check carefully about the current terms of the card and what attractions are now included or not, since my experience was so long ago and these things change. There was a similar card for Oslo that was also worthwhile. Our trip did not extend to the Norwegian fjords so some of what we did, such as Sognsvaan Lake, might be of lesser interest compared to the more urban attractions of Oslo given your limited time and that you will be seeing more spectacular scenery other places.

Copenhagen is another favorite city of mine!

Excellent suggestions from Slate, above. As she says, you will know what fits your daughter.

I was trying to think how I would have entertained "mine" as 7 year olds in Copenhagen vs Oslo. I can't add much to Slate, but a couple other thoughts on Copenhagen:

If you wanted your D to see a castle, I concur with above on Helsingor day trip to see Kronborg. The castle has kid-friendly guides and activities. Copenhagen has 3 obvious in town palace/castle options: Rosenborg, Christiansborg, and Amalienborg. I find Amalienborg the least interesting for a child of the three, but take a web look at the others to see if it is "her". They are very different. Christainsborg has options to visit the kitchen and stables. Frederiksborg Castle in Hillerød holds the Museum of National History - excellent, but wouldn't be my personal pick for a 7 year old.

Copenhagen has a number of excellent smaller art museums - all walkable and very central. I took mine to places like these at age 7, but they probably wouldn't be for all kids - especially given limited time and other options as noted above by Slate or my castle descriptions. I did visit the centrally located Natural History Museum recently - there are better options for this in the US. Conversely, the National Museum of Denmark is superb - many areas that a child might enjoy. The Viking exhibits as Slate notes and the many exhibits of historic Danish life with national costumes, furniture, toys(!), etc. If you go there, take a look at a map first and pick your exhibits. It would be easy to spend multiple days!

It is a 90 minute one way trip, but Odense is do-able as a day trip. Hans Christian Andersen venues and a "cute" historic city center. I don't think I would trade a day in Copenhagen or Helsingor for this, but you can check for your preferences.

Whatever you choose, you will not lose! (Can you add more days????)

Huge thanks for all of the information- I have some reading to do!!

I am going to try to come a day earlier and use that to add a day to Oslo if I can get flights to work!