Old Mexico is Mexico. Old York is York. But where is Old Zealand?
The first Western explorer to land on what is now known as New Zealand was a Dutchman named Abel Tasman. When he arrived in the 1640s, Tasman thought he had landed on a portion of Staten Landt, which is an island off the tip of Argentina, and he named it so. (Tasman was a little confused; it had been a long trip).
Soon after, Dutch cartographers Hendrik Brouwer and Joan Blaeu figured out that these large islands weren't actually part of South America, and Blaeu named the area Nieuw Zeeland after Zeeland, the westernmost province of the Netherlands. Zeeland is also made of islands, and its name means "sea land" in Dutch.
Englishman James Cook made three voyages to Nieuw Zeeland in the 1770s. The purpose of his original trip was to chart the path of Venus from the South Pacific, but Cook and his crew got lost and ended up in Nieuw Zeeland, which was relatively unexplored by Westerners since Tasman's original voyage. Cook charted most of the area's coastline, and he is responsible for anglicizing the name to "New Zealand."