If I were stranded on a deserted island, I would bring this 4 things that keep me safe-and-sane while living alone on a deserted island: 

1. Knife

I can use it to open coconuts, make spears for fishing and maybe crack crab legs and if the knife is big enough, like the machete, I can use it to cut woods for my shelter. 

2. First aid kit

I may get a lot of bruises and cuts while fishing and building my shelter, and an open cut could get infected very quickly in this environment, therefore a first aid kit plays an intergral role in the aforementioned cases.

3. Lighter

A lighter can keep me warm, make water potable, cook food, fend off animals and insects, and signal for rescue

4. Hammock 

Most deserted islands are filled with trees, and a hammock would provide a comfortable place to sleep those lonely nights away. It would also be above the ground and away from any nocturnal critters.