An iconic North Shore eater is Kua Aina Hawaii. It's a burger joint, but a good one at that. Burgers grilled to order and great sandwiches. I usually go for a 1/3rd lb burger with cheese, while the Mrs goes for the BLT with avocado (super crispy bacon). We're also fans with their shoestring fries, but they don't salt them, so you need to do that as soon as you get your order!
Ted's Bakery | Sunset Beach Hawaii makes the iconic Chocolate Macadamia Haupia pie and is located near Sunset Beach. The other stuff, not sure how good it is.
Of course you could hardly go to the North Shore without at least hearing about the shrimp trucks. I haven't tried them all myself, so I'll defer judgement to the pros: Big Shrimpin! Our guide to North Shore Oahu shrimp trucks and stands | Hawaii Magazine
Matsumoto Shave Ice is another North Shore icon, though I must admit I don't care for them. Matsumoto's shave ice is very sugary/syrupy... So my preference is for the newer shave ice stands that focus on more natural Uncle Clay's House of Pure Aloha | All Natural Shave Ice | Buy Local Hawaii located in Aina Hina near Waikiki. But my favorite is Ululani's on Maui.
I have yet to visit, but Kahuku Farms is a really popular North Shore destination for food and for things to see.
Your timing has to be good for this one, but another increasingly popular attraction that does have some eating options is the Haleiwa Farmers Market: One More Reason to Head to the North Shore Now - Biting Commentary - December 2014 - Honolulu, HI. It's different from the much larger and more popular Kapiolani Community College (KCC) Farmers Market.
If you're looking for something more upscale, however, your only option on the North Shore is the restaurants of Turtle Bay Resort. Though to their credit, Ola is rated well in local food publications.
For something a little more casual, Haleiwa Joe's gets good reviews on Yelp and some of my friends seem to like the place too.
Hope this helps!