Good Edinburgh stuff:
Zoo - easy bus trip from city centre. But cheap but fills a day. 10 is a great age for it.
Dynamic earth - kids went with school, bit worthy, never seemed like much cop but some folks like it
Museum on the mound (money museum) - think it's free in and interesting for a 10yo.
Castle and Holyrood - expensive and not really worth it for 10yo
Camera Obscura is great - it'll take ages to get around with all the stuff. Go for a walk up onto the esplanade, and then down the royal mile having a look at the closes then to the museum of childhood. Pizza in civerinos in Hunter square in the middle.
Free stuff
National museum - always good, try to see the clock on the hour and play in the science exhibits. The Scottish galleries are good, as are the stuff about space and animals - perfect for a 10yo
Portobello beach - I just like it, probably not that exciting if you're from somewhere with a good beach, but it's good for a walk along.
Calton hill - not as high or as busy as Arthur's seat but good views, a decent walk and you can play on the Athens of the North, as long as you're careful ( there's quite often music and general partying up there over nice weather/long weekends). Lunch afterwards in valvona and crolla on elm row.
All the Harry potter stuff - worth a look if it's your bag.
Walk down the water of Leith to the port and have lunch in the king's wark or Victor Hugo (although a 10yo might prefer Mimi's bakehouse), then get the 16 bus back into town.
You'll need to plan carefully and CHECK the tides, but cramond Island is a good trip. Bus 41 from city centre, take a lunch or walk out and back before having lunch in one of the cafes. It's only accessible at low tide and it's easy to get caught out so check before you go.
Not Edinburgh
Glasgow - science centre or transport museum are both awesome.
Stirling castle is ace but I don't see why you'd bother if you've discounted Edinburgh castle.
Wallace monument is cool and has good walk up and around it - there's a bus that works well for it.
Deep sea world in North Queensferry is good if they're still obsessed with sea life (I fucking hated octonaughts)
Dundee is a great day trip by train - good museum, lots of oor Willie/Beano /DC Thomson stuff and a really good Lebanese restaurant I can't remember the name of - starts with T. And you can walk up to the top of the hill behind the city if you're feeling enthused.
North Berwick - good day out from Edinburgh. Take the train or the bus. Good beaches, nice walks, you could try direlton or tantallon castles (tantallon is an arse ache by public transport). Sea bird centre in NB is surprisingly good. You can walk up Berwick law and see the jaw bones. All great for 10yo.