Part of the fun of renting a car and driving from Auckland to Bay of Islands (aside from occasionally checking to make sure you're on the right side of the road) is the drive itself.  The small towns along the way, like Waipu, contain hidden treasures. In Waipu's case, that's a wealth of Scottish heritage information (Waipu hosts a Highland Games festival January 1 of each year, and can become quite crowded in the high season).  You can take the detour to Kawakawa, for a pit stop - an artist named Hudertwasser designed public toilets there that have become quite an attraction.  Or you can simply take in the green rolling hills and surprise vistas.  With a stop, the drive is doable on a youngster's three-hour in-car tolerance level.  And the beaches are phenomenal.

We based in Paihia, on the east coast. Paihia features a number of places to stay (including a family-friendly hostel), and several great places to eat.  Our favorite was 35 ° South - with its built-in floor-to ceiling aquarium and over-the-water views.  We went back both nights, despite the higher price than the pizza place next door to our hotel, because dining with a one-year-old seated next to an aquarium is almost like being on a date.  

Paihia has the advantage of being two minutes from Waitangi Treaty Grounds, with its beautiful park, a full-sized whaka, and major cultural center.  It is 15 minutes to Kerikeri, where we'd hoped to kayak, but the weather and babysitter didn't cooperate. It is the base for dolphin-watching and boats departing regularly for Hole-in-the-Rock.  And there is a quick ferry (much better for her short attention span) to Russell (formerly known as Kororareka, which is approximately 'Tasty Penguin' in Maori).

Also, a short walk on the waterfront path leads you to, among other things, a boat-themed playground with swings, and an incredible view of the harbor.