The language we call Pidgin here in Hawai‘i is actually a creole language which developed on plantations. There are many creole languages spoken around the world, including Jamaican Creole, Louisiana Creole, Papiamento (Aruba, Bonaire and Curaçao), Tok Pisin (Papua New Guinea), and Haitian Creole.
In the nineteenth century in Hawai‘i, Cantonese, Hawaiian, Portuguese, and English speakers came together in the context of sugar plantations. To communicate, they developed a simplified language (called a pidgin) which used the resources they had -- vocabulary from Hawaiian and English, and very basic grammar from Hawaiian, Cantonese, and Portuguese. When large numbers of Japanese immigrated in the 1880s, Pidgin English was already the main language for communication on plantations, so the Japanese workers learned it quickly and contributed their own vocabulary items to the mix. These plantation workers then had children who grew up with Pidgin English as their main language for interethnic communication, particularly in schools. Over time, the second and third generations used this language extensively, and the result was that it developed into the fully-fledged language that linguists call Hawai‘i Creole. This language has its own distinct grammar which is entirely different from English. Though much of the vocabulary is borrowed from English, Hawaiian, and Japanese, the meanings are often quite different. So, the question is, why do we call this language Pidgin, even today? The answer is that even though Hawai‘i Creole isn’t technically a “pidgin,” the original name has simply stuck with us since early plantation days.