British Nationality could be obtained through your birth, or through the birth of any one of your parents or grandparents, in the Solomon Islands. This arises because of Britain's relationship with the Solomon Islands and Britain's Colonial history.
To determine what form of British Nationality you might have, an analysis of your family's history must be done, and then applying it to the Nationality laws of the Solomon Islands that were passed over the years.
STATUS OF THE SOLOMON ISLANDS
From 1893 until 06.07.1978, the Solomon Islands were a British Protectorate.
From 07.07.1978, they became an independent Commonwealth country.
Bougainville Island is the main island of the Autonomous Region of Bougainville of Papua New Guinea. This region is also known as Bougainville Province or the North Solomons, and includes Buka. Although Bougainville Island is geographically part of the Solomon Islands archipelago, the nation of the Solomon Islands is a separate state.
Click here for more information on the status of a British Protected Person (based on birth in a British Protectorate).
For more information on how British Nationality could be gained through your birth (or a parent's birth) in a British Protectorate coupled with a UK-born grandparent, watch this short video by Philip Gamble:
Birth in Protectorate and UK-born grandparent