Hawaiʻi Map Kauaʻi Atooi
Prior to European contact Polynesians had no written charts or maps of the islands. Navigational information was passed down through generation of skilled ocean navigators, legends, speeches, genealogies among other ways of traditional communication and preservation of knowledge in Polynesia.
The first maps of Polynesian islands including Hawai'i are inconsistent with current maps even when the names themselves remain the same. This is occurs from a few reasons. Hawai'i among other Polynesians had no written language, the Europeans that where mapping the pacific where themselves from different European countries, and the vowel sounds in Polynesian languages were foreign and difficult to write on a first voyage.
On Cook's map of what we know as the islands of Hawaiʻi were labeled as "The Sandwich Isles"
On this same map he had labeled the island of Hawaiʻi as Owhyhee , the island of Ni'ihau as Oneeheow, the island of Oʻahu as Woahoo, the island of Molokaʻi as Morotoi, the island of Lanaʻi Ranai as the island of Kaho'olawe as Tahoorwa , the island of Maui as Mowee and the island of Kauaʻi as Atooi. Interchangeable letters
Comparing the initial spelling of the island names shows that they are spelled more from pronunciation of initial contact with foreign speaking people. Some of the letter were interchangeable with other Polynesian languages such as T and K , L and R and Cook's map would suggest the use of T and R in various islands although modern Hawaiian language now accepts K and L in stead of T and R.
O and A as a prefix
Also note that the O was used as a prefix, for example O Hawai'i , O Maui , A tauai, meaning Hawai'i, Maui and Tauai.
It would then make sense that early map makers labeled Owyhee hearing O Hawaiʻi, Oneeheow hearing O Ni'ihau and Atooi hearing A Tauai.
Atooi - A Tauai - Tauai - Kauaʻi
With consideration of pronunciation and interchangeable letters the gap between Cook's map and how we spell and pronounce the Hawaiʻian starts to close. However, the Island of Kauaʻi keeps true to it's nickname as "separate kingdom" in this comparison.
On his map Capt Cook marked the island of Kauaʻi Atooi and Waimea Bay as Wymoa. Even with pronunciation Atooi is still very different from Kauaʻi. To help bridge the gap of the name of Kauaʻi and why Cook labeled it as Atooi, we should also consider other early European documentation of the island.
Recognized today as the island of Kauaʻi (often spelled Kauai), the name of the island has a few historical variations which include Cook's map Atooi along with Atauai / Akauai and Tauai / Kauai.
Kauaʻi was noted to use the T instead of the now accepted K. Kauaʻi therefore seems to be a variation of Tauai. The Island that we know today as Kauaʻi was most likely named Taua'i at the time of first European contact.
When the natives were asked what the name of the island was, they likely referred to as A Taua'i using the "A" as a prefix in the context of the native language.
Without standards for written language of the many vowel sounds of Polynesian based languages, along with the speed at which the said it, A Tauaʻi was most likely documented as Atooi from an attempt to spell out the pronunciation.