As a resident of Utah, it is seems kind of strange how connected the Mormon church is to Polynesia. What's the historical reason for this?
The LDS Church believes that Native Americans and Polynesians are all Lamanites, a tribe from the Book of Mormon:
The term Lamanite includes all Indians and Indian mixtures, such as the Polynesians, the Guatemalans, the Peruvians, as well as the Sioux, the Apache, the Mohawk, the Navajo, and others. It is a large group of great people.
They have taken an interest in preaching specifically to those people, as they feel that it is part of the covenant god made with them, that their people would be brought back to the truth. From the same article:
The Church has always maintained a tremendous interest and concern for the Indian people and all of the Lamanites. As early as 1845 a proclamation was issued by the Church, which said in part: “The sons and daughters of Zion will soon be required to devote a portion of their time in instructing the children of the forest (Indians), for they must be educated and instructed in all the arts of civil life, as well as in the gospel. They must be clothed, fed, and instructed in the principles and practice of virtue, modesty … customs, dress, music and all other things which are calculated in their nature to refine, purify, exalt, and glorify them as the sons and daughters of the royal house of Israel and of Joseph, who are making ready for the coming of the bridegroom.”
Because of that, the LDS Church sent missionaries there as part of the initial Christian missionary wave into that region of the world. Since they were first in, they had a lot of initial converts, cementing their status as a major denomination in the region.