Water purifier, fishing rod, First-aid kit and a knife
The knife: It's a fantastic crafting tool to make other weapons out of wood or stone that you may find, as well as being a killer to any unfriendly inhabitants of the island.
Water purifier: If you don't bring this then good luck surviving, idiota ;) Probably the most vital thing on the list, without this you cannot drink any of the tempting sea water surrounding you.
Fishing rod: Food + something to do = survival
First-aid kit: In case i stub my toe, trip over a vine, fall into a pit or get attacked my a wild blowfish >.<