I really would like you to ask again in the Mega-Thread, because then we can build up a wee empire of knowledge (heh, a taonga) of our own about New Zealand. But I can answer here as well!

The answer is totally yes. The Maori have some really amazing Oral History narratives about their arrival in New Zealand. The narratives impact how they structure their societies to the present day, especially with the concept of mana being able to be inherited ((talked about here)).

The first thing to remember is that Maori Iwi are incredibly fractured; what one tribe says another might not. Anthropologist Dame Anne Selmond actually goes further in seeing that for Maori, it doesn't matter if the particular story they've been handed is correct - its value isn't that it's true or false like Western knowledge is, but in that it has been inherited from the ancestors, and will be handed on in turn. Problematically, it also means... why would you share a treasure - a treasure your ancestors gave you - to some outsider looking to publish it and dissect it? Often Kamatua (elders) of Maori Iwi simply refuse to tell these stories even within the tribe, and there are serious ethical pitfalls to negotiate before a researcher can continue.

From what we know, however, the biggest shared story - with Europeans, that is - is definitely... and I see /u/rraarr has totally beaten me there again! =D The story of Kupe, legendary discoverer or New Zealand. Those stories identify his origin as an island called "Hawaiki". As a story, most New Zealanders will know this one. In the "biggest" variant of the story he was fleeing Hawaiki after committing a pretty serious murder and running off with the murderee's wife. Bearing in mind this was the story taught in New Zealand schools as the Maori origin myth right up until the 1980s. How common the story was in actual Maori practice is a really hard question to answer.

I'd like to emphases again this is by no means the only answer; Maori tribes have an astonishingly rich cultural heritage in these stories. Another constant in these stories, however, is tracing back to what canoe the particular tribe arrived on. Some tribes claim there were only seven canoes. The names of these are the most famous in New Zealand, and tribes from a more senior canoe will claim mana over those from a lessor. But stories give dozens of names for these canoes, and it is impossible to reconstruct if they all arrived at the same time (although this is seriously unlikely), if some of the names are corruptions of others, or even when they arrived.

The earliest archaeological material we've got is tentatively dated from about the 13th Century, although the oral tradition would suggest as early as 900 AD, and some historians have disputed the ages of the findings and suggest sometime after 1400 AD is more likely. Conservatively the Maori have been in New Zealand 600 years, and that's a long time to hold an oral history tradition completely intact.

I am not sure if this helped with your question or just made you confused, but it's a complex question that I could talk about for ages. =D

You should ask this in the mega thread :) http://www.reddit.com/r/AskHistorians/comments/1x4mnd/waitangi_day_new_zealand_history_megathread/

Ok, I agree with /u/TasfromTAS because /u/CrossyNZ's answers are amazing - you definitely want to get his input. But in the spirit of pointing you in the right direction:

In the oral tradition, Kupe was said to to be the first to sight NZ, and it was the cry of his wife ("He ao! He ao!" / A cloud! A cloud!) that gave our land the name Land of the Long White Cloud / Aotearoa. Kupe sailed back to Hawaiki and lead the great migration to New Zealand. Each waka (canoe) is the basis of the iwi/tribes - like Arawa, Takitimu, Mātaatua, Tainui, Tokomaru etc.

Anyway, that's my basic knowledge!

edit: and from what I understand, Maori arrival in New Zealand is dated to around 950AD.