What sort of partnership visa are you going for yourself? Residence, or partner-sponsored temporary work visa?
At first glance, I think yes, your son will need a student visa. He wouldn't qualify as a dependent child, as he's not the child of your partner, unless he has adopted him formally, and you yourself aren't yet a citizen or resident.
Dependent child – you are a dependent child with a parent who is a New Zealand citizen or resident.
http://www.immigration.govt.nz/migra...categories.htm But if you are applying for residence under the Family Stream, it may be worth asking INZ if you can, at the same time, apply for a dependant child visa, to be processed along with your case. (This procedure happens when one partner is applying for a work visa, and the other is allowed to apply for a partner-sponsored work visa at the same time, with a letter linking the two files. We on here are nothing official, so I'm only guessing that this MIGHT be allowed, but I think it's worth asking.)