According to The US Board of Geographic Names, the name of the state remains Hawaii (unchanged since it was made a state) but since 1999 the name of the "Big Island" is now Hawai'i, preferred by ethnic Hawaiians.
I was having trouble with some of the pronunciations when we were there, and a woman told me to pronounce the pause as in "uh-oh". No probs after that.
Two reasons. First because the island of Hawaii is commonly known by its nickname, the Big Island, whereas the rest of the islands are known by their actual names. And second because Honolulu, the capital and (by far) the largest city, is located on Oahu.
In Hawai'i, we called the Island of Hawaii "Big Island" and not "Hawaii" so my first instinct was to put "Big Island" and I really think you should make that an answer
Pack your bags, we're leaving