I guess that the question is not so simple. I think that a traditional Hawaiian song would have come from the true Hawaiian culture, before the United States annexed Hawaii, or at least have been written by someone of the Hawaiian culture.
... I think that uke1950 got it right on the 3rd post, but it is good to know what the ukulele playing public and especially native Hawaiians think is a traditional Hawaiian song.
You're right, given Hawai'i's history it isn't a simple question at all! But it's one I find really interesting and I've really enjoyed reading the responses here.
I'm not going to claim even remotely to be an expert here, but many of the songs the ukulele-playing public that I encounter consider "traditional" Hawaiian songs are really Hapa Haole songs, often written by Mainlanders to exploit the tourist interest in Hawaii that was rampant during the first half of the last century. On first glance, I'd definitely *not* include those on the traditional list...
...except it gets even more complicated. "Pearly Shells" is a good example - the well-known English lyrics were written by Webley Edwards, a mainlander who moved to Hawaii in the 1920s and had a popular radio show there for years. But the original song is much older and the Hawaiian version is much older. So is it "traditional"? One could say both yes, and no.
For that matter, some of the songs that can be closely associated with Hawai'i - such as
Aloha ʻOe by Queen Liliʻuokalani and
Hawaiʻi Ponoʻī by King Kalākaua - could probably not be considered "traditional Hawaiian" in that the royal siblings were educated in missionary schools and exposed to western classical traditions.
And I don't know if it's accurate or not, but years ago I asked a teacher what "traditional" Hawaiian music was, and his reply was "chants." If I remember correctly, he said that prior to Western influence, Hawai'i didn't have much in the way of musical instruments beyond percussion, and what we think of as Hawaiian music is really a blend of many cultural influences. If I'm not mistaken, this is also touched on in John King and Jim Tranquada's excellent history. (
Jim - if you're reading, please give us a few words!)
So perhaps where this gets sticky is the use of the word "traditional." I'd be inclined to agree with
Kekani's response, but once you start browsing that link you'll notice a bit of hapa haole and more contemporary music mixed in. The lines get blurred!