Papatowai Beach is a nice, relatively secluded beach on the East Coast of the South Island. This beach has a picnic area on the banks of the beach, and you can head down little beach paths through the beach grasses to access the beach. A stream runs onto the beach and out to sea and would be good for little children to play in. The sands are a dark golden colour and this would be the perfect beach for frisbee or just to relax on during a busy day. The beach is surrounded by native New Zelaand forest and there are a few short walks that you can go through in the area. I personally have not done these but would really like to one day. I am sure they would be quite easy and would give you nice coastal views.
There is a small coastal community nearby that you would be able to get accommodation within. This may be a good idea as there are long stretches of road between Papatowai and the nearby towns.