Depends on what sort of holiday you are after. A touring one would be easy enough because you can always stop at a park/river, in the North Island especially you would never go more than an hour before coming to the next town.

If you are wanting just to go to a few places and have a good look around at each then I would suggest bookabach for accomodation - about the same price (or cheaper) than motels but it is someones holiday house so you have bedrooms/yard etc, much more comfortable with a toddler imo.

If you aren't planning to hire a vehicle, you will need to stick to the main centers - smaller places just don't have public transport.

I live in the volcanic central plateau, which is the center of the North Island. Activities around here are mostly outdoor things, we go on bushwalks, to the river and boating on the lake.
Taupo/Rotorua area is good for hotpools/mudpools.
Dunedin rocks, there are aquariums, a castle, albatross colonies. It is a beautiful place, but if you go there then make it Jan or EARLY Feb, because by the end of Feb all the students are there for Orientation week.
Central Otago including Queenstown/Wanaka/Cromwell is lovely. Keep in mind that Qtown is for tourists so expect to pay more if you are there (even Macca's costs more!) All three have lakes and there are touristy things like the luge, gondola rides, puzzle world, quite a bit of stuff to do.

Really wherever you go is beautiful and there are things to do - with a 2.5 year old as long as they can get out and move around then they will enjoy it.