For this, I mean such materials as obsidian or other materials that fracture into sharp edges easily, or can be fashioned into them easily. I'm trying to come up with basic tools for the local tribal population, and I figured the first place to start would be what materials are available to fashion those tools out of. Flora and fauna are both plentiful (this is a world where speculative "prehistoric" life, such as imaginary derived species of pliosaurs, exist, so bone/scale/hide/etc materials from fauna and wood materials from trees would be easily accessible).
Copy the indigineous tech of a place like your place.
You can pick a people you think are like your people. Or if you want to start fresh, pick a place that you think is like your place. Then copy the tech of the people who lived there. Archipelagos come to exist for different reasons. Tech using stone would be dictated by the geology of that place and available stone.
Stone age craftsmen in New Guinea used obsidian. But Hawaiians lived on volcanic islands where obsidian is rare. They made tools out of basalt, as well as shells, bones, and teeth. Tonga is a coral atoll and the available stone is limestone - so they traded for tools made of better stone. You could have a people like the Inuit who make some iron tools using meteoric iron. You can mix and match if you like, moving the stone age tech of inland Africa to your islands and asserting that the geology of the place allowed it.
The more you can lift from real things, the more your own inventions will sparkle when you set them in a background of things that are real.
If you are looking for cutting edges, you can either go for stone (flint or obsidian), or, in case the geology of your archipelago doesn't support it, shark teeth can be a valuable substitute. You are in an archipelago, at the end, so try to use the sea as much as possible.
I think it is mentioned in Melville's Typee that the indigenous of Polynesia used shark teeth to shave their head. Sure it is not like the modern 25 blades and football player included safety razors, but it's one of the best material you can find.