I'm not sure either of these count a strictly "Hawaiian" cuisine but I highly recommend both.

Royal Thai Cafe in the Keauhou shopping center on Alii Drive. Pineapple fried rice was very clever and service was delightful. Definitely my top choice on the island.

The Gallery Reestaurant at the Mauna Lani resort. It's basically the clubhouse for the resort's golf course. We had lunch there twice during a week's stay in March and found the food to be consistently delicious, the service very friendly and efficient, and the atmosphere (open air, overlooking course and ocean) very relaxing. We went there at first just as a convenience (we were staying at the resort) but were very pleasantly surprised and regretted each time we skipped it in favor of other "recommended" places. Definitely need a car to get there from Kona town.

Cafe Pesto in Kawaihae was highly recommended in various guidebooks but was a colossal disappointment. Very mediocre food and slow, sullen service.

FWIW, two other places on the Big Island I'd highly recommend, although both somewhat distant from Kona:

Restaurant at the Kilauea Lodge in Volcano. Wonderful dinner and service capped off a sunset lava viewing in the nearby Vocanoes National Park. The atmosphere was a bit more formal than most places on the island, and we were pretty grubby from our hiking, but were made to feel very welcome nonetheless.

Kuhio Grille 1 in Hilo. Non-descript storefront in a shopping center with a Sears and a Hilo Hattie's. Probably the closest thing to traditional "Hawaiian" food in this list. Again, very friendly service.