The main issue here is that the economic development needs to be sustainable and it should benefit everyone. Historically, the problem with throwing development money at such places is that it can lead to resource degradation and also making a small kleptocracy rich while actually worsening the lives of the average islander.
The first step would be to establish a school and reliable internet infrastructure, including advanced e-learning technology so that islanders could stay in their communities while getting a solid university education.
Next would be to look at what sustainable economic activities were possible. Exports of hand made craft items is sustainable and benefits local small businesses if correctly managed. Limited amounts of high-end tourism with profits going into a sovereign wealth fund would work if (1) it wasn't just a matter of a few foreigners and local power brokers getting rich and most people getting deprived of land and exploited (2) the sovereign wealth fund being administered transparently and honestly (which is usually, alas, quite rare. Developing e-businesses might also be a good possibility.
First, we have to understand that this may simply not be possible. There are places in the world that lack the sorts of advantages that can allow them to develop economically. I grew up on an island in Micronesia that sounds like this hypothetical (except that it had about 18,000 people or so). The island and others like it continue to struggle to find a way to economic growth.
Second, we must understand that education is not a panacea. If you have an island full of people with college degrees, it will not change the fact that there are few possibilities for a solid economic foundation for the island. We see this even in small rural communities in the United States. Good jobs that make full use of educated people tend to cluster in cities where they catalyze one another. This will never happen on an island of 3,000 people.
Third, handicrafts and high-end tourism are not likely to make an economic foundation either. Handicrafts do not sell in the bulk or at the prices that allow them to be profit centers. Tourism provides mainly menial jobs that have no potential for growth.
In short, there is likely nothing that can be done to make an island like this wealthy. Its best hope is to subsist at a level that is many rungs below that of rich countries like the US. Some islanders will leave for jobs in rich countries and send remittances. Handicrafts and tourism will bring in some money. Fishing and forest resources can bring in some money but must be managed so they will not get stripped the way Nauru's phosphate was stripped. People will need to be encouraged to take up sustainable farming that can feed local needs so that the locals do not depend on imported foods. Perhaps grants can be obtained for things like solar power so that there is less dependence on oil that needs to be brought in. The island will need to reduce its consumption of outside products as much as possible and make the best possible use of its few resources. But it will never become wealthy in comparison to countries like the US.