Bear in mind that I don't live in ChCh, so I don't know where the suburbs are (and, most particularly, don't know how they sit with the earthquake damage), but a quick look on RealEstate (http://www.realestate.co.nz/ - this is a combined rental and buying website for general estate agent advertising - i.e. it isn't run by one particular company, and, along with TradeMe (which you can browse but can't register for until you have a NZ address), is the best place to go online to look for places) suggests between $200 - $500 per week (the $500 per week is the 'Executive' end!) The majority of them seem to be between $200 and $300 per week, depending on area and whether or not they are furnished.
You will need 3 weeks rent for a deposit as well as a week in advance (so, similar to renting in the UK).
Do bear in mind that when properties are let unfurnished, that also includes no fridge/freezer or washing machine. You should get an oven. (I got caught out by this and suddenly had to go shopping for whiteware when we didn't really have it in the budget!)
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