In 1856, the residents of the small atoll of Sikaiana (also called the Stewart Islands) offered sovereignty of their land to King Kamehameha IV of Hawaii.
As we know, Hawaii was later annexed by the United States. But OTL, this threw Sikaiana into a hazy area legally. Neither Hawaii nor the US ever ceded Sikaiana to any of the colonial powers in the Solomons, which by international law could leave the island as an American territory (or part of the state of Hawaii), although sadly, the petitions of the islanders to the Hawaiian government have been ignored thus far (and the Solomon Islands has effective control there).
But what if the United States not only took up Hawaii's claim, but used it as a basis for expanding further into the Solomons? Here they'd be competing with Germany and Britain over the colonisation of the territory. This is obviously going to involve a lot of negotiating between the three nations. Perhaps the United States abandons Samoa and accepts the British claim in the Alaska boundary dispute in exchange for what was the British South Solomons (the southern few islands including Guadalcanal, San Cristobal, and Malaita). Or even a three way partition of the Solomons, with the US simply gaining control over Malaita and the nearby islands (which is the nearest land to Sikaiana), Britain gaining the rest of the South Solomons, and Germany gaining the North Solomons. I'm not sure it would be possible for the US to gain the entire Solomon Islands (which would include Bougainville), outside of some German-American War or if the US joins WWI from day one and decides to take the islands.
So what are the islands like in the modern age? I'm guessing somewhat wealthier than the OTL Solomon Islands, but still with living standards far beneath the rest of America. The territory will suffer similar issues to other territories of the United States--disproportionate amounts of corruption, unemployment, and poverty. With a population of around 300,000 (although the entire Solomons plus Bougainville might make a better argument, although given Puerto Rico's issues, it probably won't happen), it would probably never seriously be considered for statehood.
Culturally, I suspect it would be a bit like American Samoa. The Solomons could produce a large amount of talented NFL players.
Thoughts on this?