In Quentin Tarantino's The Hateful Eight, there was a character who says she's from Auckland, New Zealand. Given the timeframe of the movie being several years after the Civil War what potential motivations would someone from New Zealand have come all the way to the United States?

If we take the 1870s as the time period for the setting of The Hateful Eight, we can also pinpoint certain contexts occurring in New Zealand at the time which might give us clues as to why this character was present in America.

The first reason, is trade. Ships regularly plied their way between the west coast of North America and New Zealand, starting several decades before the 1870s. One of the wealthiest businessmen in New Zealand, Thomas Russell, made a fortune during the goldrush by selling shiploads of onions to San Francisco, seeing that the large number of new arrivals to the city would require food. His father and brothers had travelled to California themselves, hoping to make their fortunes on the goldfields. There is plenty of evidence to suggest regular travel between the two regions was possible.

A major cause of migration during the 1800s was a desire to make one's fortune. New Zealand had been populated by British settlers who were drawn there through advertising campaigns in Britain. These were largely false or embellished in order to encourage more migration. They had to be very enticing, as New Zealand was so far away when compared to the other English speaking destinations available: Australia, Canada, America. But not everyone who made it to New Zealand was successful, and to that end, it's possible that a number of people, upon realising that New Zealand didn't offer the opportunities they hoped for, instead chose to move on to the other colonies. Australia and California would be major destinations here if a return to the UK wasn't an option. This is especially likely because at this period, New Zealand was still developing from provinces into a country, whereas America was far more formed as a state. (Edited note: I forgot to mention this when I first wrote it, but during the 1870s/80s/90s, Auckland, the biggest city, went through several economic fluctuations including a bust that lasted from the late 70s to the early 90s. This could have been a problem for a lot of people's livelihoods).

The final reason is related to opportunities, but of a different sort. 1872 marked the end of the New Zealand Wars, with the peace being made between Te Kooti and the government. During the wars, there were examples of mercenaries arriving to fight in New Zealand (Von Tempsky, a Prussian soldier who had also served in Central America being the most famous of these), and the frontier battles were coming to an end in New Zealand. The West in America was probably seen as a land of opportunity, and as a fellow English speaking colony, New Zealand would have been a prime market for advertising from migration companies (I don't have any proof of this, it's just an idea). Further to that, the image of the West as a wild frontier would be a drawcard for the soldiers who were now being released from service to peaceful lives. Many of them weren't actually New Zealanders, but men who had come from abroad in response to government advertising. The government offered them bad quality land, so many did their service, sold the land, and left again. The likelihood of mercenary soldiers who had been drawn from Australia to New Zealand then deciding to move on to entirely new pastures where there particular skills might be useful makes America a drawcard.

Wikipedia lists several gold rushes through the 1860s/70s, and those drew people from around the globe who were forever pursuing the wealth that gold symbolised.

I don't know whether any of this would work for QT's character, or if he just chose a random nation (or maybe it just fit the actress' accent!), but there was definitely historical precedent for travel between New Zealand and America, and several valid reasons for someone to make the trip.