The Kingdom of Kaua'i
(alternate and still surviving Kingdom of Hawai'i)
Simply put, this comes from a TL where Kamehameha the Great died early in his life-long campaign of unifying the Kanaki of the Hawaiian islands. Prior to that, James Cook never made a second landing on the islands in early 1779 and the islands went under the radar of Europeans for a few years longer than in OTL. Before the end of the 18th century, the French colonised the southern islands, with the British following suit. Thus, the Sandwich (OTL Hawaiian) islands were separated into two different national colonies. The French backed the Hawaiian chieftain dynasties, the British eventually sided with Kauai'an rulers. The French put down a southern Kanaki revolt in the late 1840s and marginalized the natives' political power, the Brits and northern Kanaki had better luck and House Kaua'i gradually took control over the entire English-speaking part of the islands. Due to problems at home, the French had to give up some of the harder managable colonies (well, make a deal wit the Brits about swaping some). Thus, the southern portion went into British hands as well, and with the local southern aristocracy nearly gone, House Kaua'i had little trouble uniting the entire archipelago.
Thus, the Kingdom of Kaua'i was established in 1892. It later gained autonomy in 1927 and full independence in the early 60s.
And since many of you have complained about the lack of deeper backstory, I'll post the whole mini TL behind this in the following post.
Some explanation behind the symbolism in the flag :
Blue means sea, green means Hawaian vegetation (especially the lush Kauaian one). The leaf fan at the center and the war paddles by its sides are traditional in Polynesian heraldry and were used on an
early unofficial OTL flag of the Hawaian monarchy. The central figure is the mahiole (headdress/crown) of king Kaumualii, the most famous member of House Kaua'i, the founder of the unification effort spearheaded by the British-favoured Kaua'ian Kanaki. To the right is the cloak of the young Hawa'ian ruler Kīwala’ō, who defeated Kamehameha ITTL (it was the opposite OTL), securing him the rights to rule over Hawa'i. (The French showed up a year or two later and he started working on trading and allying with them, putting the conquest north of Mau'i aside for a few years - unfortunately, the Brits took the northern islands by then...) The presence of the cloak near the mahiole is supposed to symbolize the dynastic and ethnic bond established by bringing the Hawai'i and Kaua'i dynasties together during the founding of the kingdom in 1892. The star is also symbolic of the union - the eight points representing the eight main islands of the archipelago.
Incidentally, before unification and during British rule, the Kaua'ians used a flag with a blue background, a British flag in the canton and the Kaua'ian mahiole to the right of the canton, on the aforementioned blue field.