My question, as well. There is a big difference among those environments.
just for fun, let me jump in and suggest an actual island that has been deserted during historic times - San Nicolas Island, off the coast of Southern California. It currently hosts the US Navy, whose biggest issue today would be a beverage shortage in the PX.
It is renowned as the location of the 'Lone Woman of San Nicolas Island," who lived alone on the island for 18 years during the 19th century. AFAIK, she had none of the items available on our proposed list.
San Nicolas does have some permanent springs, productive tide pools, and edible native plants. Much better climate than those frigid, windswept Alaskan Islands....
In that case:
Record lows are above freezing, but not by much, so shelter is a must, but could be improvised with local materials or an improved cave.
Fresh water sources, so the water filter goes down on the list.
Much of the available food can be eaten without cooking, but cooking would still be preferable.
A tarp is very versatile. Signaling can be done with local materials and fishing can be done a variety of ways.
So my choices:
Tarp
Pot
Rope
Knife
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