This is a very interesting question that was asked by many scientists – what does Maori lore tell us about the now extinct megafauna of New Zealand? The results of all previous investigations are rather sobering, all so-called traditional accounts seem to date to the time following the arrival of the Europeans in New Zealand.
I want to mention only one of them here.
The first account dates from the middle of the 19th century.:
“The natives speak of another member of this family, which they name the kiwi papa whenua, a still larger species, which they describe as having been full seven feet high; it likewise had a very long bill, with which it made large holes in the ground, in search after worms. This bird is now extinct, but there are persons living who have seen it. Rauparaha told me he had eaten it in his youth, which might be about seventy years ago [ca. 1785], and when that Chief died, his corpse was said to have been ornamented with some of its feathers.” 
This second account refers to the first one and was made just ten years later.:
“Kiwi Papa Whenua. Seven feet [ca. 2 m] high. One of the last birds to disappear. There are still men who have hunted it.” 
The Kiwi papa whenua accounts may indeed refer to one of the smaller or middle-sized moa species, one that was about 2 m tall and that may have survived longer than most of the other moa species, but probably not into the early- or middle 18th century; it might thus be referring to the so-called Upland Moa (Megalapteryx didinus (Owen)), a species that officially died out around 1500 AD.. However, when reading the first account, it is very clear that this description has been mixed with that of a typical kiwi, thus it is quite clear that these accounts are no eyewitness reports.
The term Kiwi papa whenua might be translated as ‘Ground kiwi’ or maybe ‘Kiwi of the land’ which is not very meaningful. It is furthermore rather unlikely that the Maori would have connected the diurnal, rather large, long-necked moa species with the completely distinct kiwi(s), thus it is very unlikely that the term ‘kiwi’ would have been used for any of these species.
Nevertheless, such old accounts remain very interesting, and I will go on posting more of them in the future.
 Richard Taylor: Te Ika a Maui: or, New Zealand and its inhabitants, illustrating the origin, manners, customs, mythology, religion, rites, songs, proverbs, fables, and language of the natives: together with the geology, natural history, productions, and climate of the country; its state as regards Christianity; sketches of the principal chiefs, and their present position; with a map and numerous illustrations. London: Wertheim and Macintosh, 24, Paternoster-Row. 1855
 J. B. Ellman: Brief Notes on the Birds of New Zealand. The Zoologist 19: 7464-7473. 1861