Oddly enough, we may not have Hawai'inui in our stories (at least none that I'm aware of), there is Hawai'iloa, a navigator who settled on the island of Hawai'i and they say it was named after him.
Well we have Hawaiki Roa in our history, but that was the name of an island, not of a navigator as far as I'm aware. Maybe a connection? Or maybe a false friend?
The story I'm looking at now mentions Hawaiiloa the chief mentions how he was active in his fishing excursions, sometimes gone for mos. during which he'd roam about the ocean in large waa. "The Big Island" which was an island he landed on found to be very fertile and he named it after himself. He and his people lived there for a long time and returned to their native country intending to come back to Hawaii nei (the Hawn. islands) which they preferred to their own country. He and the other travelers such as navigators, steersmen, shipbuilders, etc. and got their wives and children and others and went back to Hawaiinei. It was because of this that Hawaiiloa was called the special progenitor of this nation. Hawaiiloa called the islands after the names of his children.
After he had spent time in Hawaiinei, he made another voyage to find his brothers and their families. He found one brother and both he and the brother sailed (further?) southward and found an uninhabited island which Hawaiiloa called after his own name and other smaller island which he called after his daughter "Oahu".
Ok, so that last paragraph, dunno really b/c they say that Oahu was a chief, son of Lua and Papa. The story wasn't specific after he found his brother Ki as to where exactly he found him, and then it mentions they went south. But where the origin point was, dunno. But I can see why other places could be named Havaii.
Niue's first ancestor was called Langiatea. I'm not suggesting that Tahitians sailed to Niue or that Niueans colonized Tahiti, but I find the idea interesting that their source or ancestor is Langiatea while ours is an island called Rangiatea (which is more properly known as Hawaiki Nui).
And these are all important to know. Even something as simple as these names, these cultural tidbids give you insight into the language, the change of sounds from one polynesian language to another or even in Hawaiian alone too.