According to the Baidu entry and a Wikipedia entry, Taiwan is a transliteration of an Indigenous name. Respectively:

Baidu

大员台湾古称，是由台湾南部平埔族台窝湾社（Teyowan）之名转化而来 The name Dayuan Taiwan originates from the Teyowan settlement of the Pingpu people in the south of Taiwan.

Wiki

Many scholars propose that the name of the island Taiwan actually came from the indigenous people's name, as the pronunciation of Taivoan is similar to Tayovan, the people that the Dutch met around the coast of Anping or the bay around Anping, which later became the name Taiwan. In addition, the Taivoan established a settlement called Taiouwang, which is the only indigenous community residing there whose name resembles Taiwan.[4][5]

As for other Indigenous groups' names for the island, or what the Taivoan people call the island itself I am unsure. I'm slow at reading Chinese so am just doing my best to search and translate.