Why did the Japanese migrate to Hawaii in the 1800s
Geographers talk about "push" and "pull" factors that cause migration. There were both in this case.
In terms of the "pull" factors that made Japanese want to go to Hawaii, the main one was the availability of jobs. In the 1800s, the Hawaiian sugar industry was booming and the plantation owners needed workers. Asia was the nearest source of workers and many Asians (Japanese included) went to Hawaii.
In terms of "push" factors, the main one was overpopulation and the lack of economic opportunity that went along with it. Even though the reforms of the Meiji era were taking place, Japan's economy was not yet doing so well and there were not that many ways for people to make money in Japan.