My favorite cheap activities with kids (but I don't know what is available in Christchurch these days):
library
swimming pool
beach
walking track (look on regional council and Department of Conservation websites for nearby suggestions)
playgrounds (on the weekend it's fine to go to a school's playground)
train rides
going to a cafe for hot cocoa
exploring (it's new for everyone!)
U-pick berries or apples
Cooking together (even if it's just assembling sandwiches)
We rented a car on weekends before we bought one so the exploring could go further. But we also took buses rather long distances to learn about our area. If I lived in the Christchurch area, I think Akaroa would be a top family explore for us, we have stayed out there before and loved it (also had U-pick peaches on the way).