Knife - the single most important tool.

Tent - immediately solves the shelter problem, which is more urgent than water or food. Protects against sun, rain and insects, helping to get a good night's sleep. It also is made of a great deal of material and cordage, which could be used for improvising various things if necessary. In my experience, tents collect dew overnight, and if that works on the island it's a water source. It would also collect rain water. I don't see why anyone would pick a tarp when they could have a tent.

Matches - if there's one thing Naked and Afraid has taught me, it's that without a proper fire-starter you're stuffed. Even if you know bow-drills, getting one to work on a strange island where you not familiar with the local wood is hard. If you don't get fire quickly, chances are you'll be too tired and dehydrated, or sick from drinking bad water, to get it at all.

Pot - needed to purify water by boiling. Could also be used to carry fire around. This is the choice I'm least sure of.

I was actually quite tempted by the sunscreen and insect repellent. Sunburn can be debilitating, and insects can make sleep almost impossible. However, I figure the tent will deal with both. First aid kit could be useful, but I don't know what's in it. Few kits have potassium permanganate, for example. Boots: but I'll already have footwear, won't I? A mirror would be useful for signalling, but hopefully a fire would work instead.
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Quality is addictive.