Well stuff to do in aberdeen for the tweens:
Science Centre
Cordonas (amusement park with bowling, trampoline park)
Maritime Museum
Old Tollbooth Museum
Art Gallery (to be reopened soon)
His Majesty's Theatre has a fair bit going on so a panto every Christmas, Scottish National Opera/Ballet
The Beach - easily accessible lots of little cafes on it, a nice spot for an afternoon out
Swimming pool with a flume by the beach
Ice rink by the beach
Duthie park - great play equipment, indoor botanical gardens with turtles
Hazelhead park - again lots of play equipment, cafe, maze, petting zoo
Vue and Cineworld cinemas
As for activities there are plenty of ballet, brownie, scout, swimming classes etc.
Aberdeen is great for easy access to the countryside with plenty of castles near by and opportunities for dolphin watching.
Dundee is only about an hour by train so can make a great day out for V+A, discovery, etc.
I think Aberdeen is a great place to be a tween, big enough to have stuff to do but not so big as to be overwhelmingly huge/spread out.
One note of caution - Aberdeen is at the end of the universe. It is a good 2.5 hours drive down to Edinburgh so this adds significantly to any journey south you want to make.