Well, this has more handwaving than the Queen's coronation, but I'll have a go.
Japan decides not to claim Germany's territories in New Guinea after World War One in the hopes of getting more concrete political trade-offs elsewhere.
The mandates are divided between Australia and New Zealand; Australia takes New Guinea, the Marianas and the Caroline Islands with New Zealand holding the Solomons, the Marshall Islands and Samoa.
OTL there were several New Zealand politicians with delusions of a mini-Empire in the South Pacific, but realistically there was little to gain.
In the 20s', Britain decides to turn over the administration of Fiji to Wellington as a method of cutting costs whilst feeding the young dominion's ego; Whitehall knows that in any conceivable emergency that requires the use of Fiji as a base New Zealand won't be charging rent. The Gilbert and Elice Islands (Kiribati and Tuvalu) are also given away, and Britain decides it will transfer its share of the rule of Vanuatu to to New Zealand as well.
After much discussion, it is also agreed that the Kingdom of Tonga will enter a state of 'Friendship and Free Association' with New Zealand.
Now, we could claim that New Zealand's leader is an Emperor given that the British monarchs claimed the title of 'Emperor of India' at this time IIRC. However, if we are at home to bats of the spacious Alien variety, we can have a bit of fun.
Wellington decides that it will try and co-opt the chiefs of their island holdings by offering their children scholarships to New Zealand universities. It is thought that a western education will make them see the benefits of being ruled from Wellington.
This takes time, however, and dissent continues to grow. After anti-colonial demonstrations in Samoa in 1929 are put down violently, (as in OTL) a wave of discontent sweeps across New Zealand's pacific holdings. Local Polynesian, Macronesian and Micronesian leaders feel disenfranchised and demand more respect from the colonial administrators.
By the 1930s, enough of their children have journeyed to New Zealand and back to gain an understanding of the ruling nation's culture. In their time there, they are exposed to the Kingitanga movement there.
In the early 1930s, the New Zealand government bungles the handling of protesters twice more; in Tonga several people are shot by panicked police, and in Fiji a crowd riots outside government house in Suva.
Concerned, London directs the dominion to reform its rule or face being stripped of its possessions- however minor these incidents, it embarrasses London to have this kind of situation in what is meant to be the peaceful backwaters of the Empire.
Eventually, at the Apia Conference of 1936, delegates from the island chiefs and the Kingitanga movement agree to a Pact of Friendship and Cooperation. The Maori King is the only Native leader with official representation in Wellington (in the form of the Maori seats.)
Therefore, the King of Tonga and the Paramount chiefs of the various islands agree to pledge their support to King Koroki Mahuta, as the spokesman of the native populations of New Zealand's holdings.
The popular press begin to refer to the Maori king as the Polynesian Emperor. Though incorrect- his subjects are more like allies, and many of them are not Polynesian, the term catches on.
Michael Joseph Savage's new government recognizes the chance for a compromise that will please London, the Chiefs and a New Zealand public that wants the prestige of foreign holdings without the assorted headaches.
Representatives of the various islands are sent to the King's court, though in practice they act as ambassadors in Wellington. The growth of an 'Imperial Maori' identity is remarkably rapid; scholars later assess it as arising out of the depressed economy which highlighted Pacific and Maori disenchantment with the status quo. With Savage's government coming to power in a New Zealand that was nervous about government heavy handedness following the divisive West Coast mining strikes and the bloody bungling of the protests in the Pacific empire, the Labour party was uncomfortable about taking firm action against the new movement especially when it had the economy to salvage and a social welfare scheme to implement.
In World War II, a Maori public that feels responsible to their island brothers volunteers en masse for the military after the Japanese occupation of most of New Zealand's islands. The Second Maori Battalion is the only Commonwealth force that joins the US Marines at the Battle of Tarawa; though it is a logistical headache to integrate them into an American force, the allies find it makes great propaganda to to have the Pacific Islanders be liberated by their 'Emperor.'
The rule of the 'Emperor' actually outlasts that of New Zealand's official presence in many of the islands; following the independence of Samoa et al the pretense of the Maori King's rule allows for a convenient forum for Pacific issues to be discussed. Under Norman Kirk's administration, it is successfully argued that as 'subjects' of the Maori King the populace of the Cook Islands and the remaining pacific territories deserve an additional Maori seat. In 1987 the ongoing diplomatic dispute between France and New Zealand takes another turn for the worse when many Tahitian leaders formally send delegates to the Waikato court. Even into the new 21st century, it is an accepted form of protest of Pacific leaders to make their complaints to the supposed lord of the Pacific- in 2008 both Wellington and the Waikato were greatly embarrassed when a claimant to the Hawaiian throne demanded that the Pacific chiefs recognise their 'oppressed brothers and sisters.'
Today, the 'Emperor of New Zealand' as he is popularly known is a statesman of the Pacific; Australia and New Zealand are now happy to use Imperial authority for their own ends in the Pacific. Samoan, Maori and Tongan troops formed an 'Imperial Peacekeeping' brigade that kept order while the Suva talks of 2006 finally resolved Fiji's ongoing cycle of coups; there is talk of sending the Imperial Peacekeepers to the Solomons following their success in Fiji.
Alright, it's ASB but I'm pretty happy with it for an effort while I'm tired.