The obvious dumb classical algorithm for simulating a quantum computer with n qubits runs in O(2^n) time and space. Since 2^log(n) = n is polynomial it follows that "log-space" quantum circuits, I.e. quantum circuits with a logarithmic number of qubits are efficiently classically simulable. These log-space circuits are an obvious "island" in the metaphor of the article and adding more qubits is the "quantum resource which let's you swim in the sea".

Another obvious example is what you get if there is no superpositions at any time. I.e. at every step in the computation the state is a computational basis state. Since we label computational basis states with binary strings, with a little checking you can see that circuits with no superpositions are exactly the same as reversible classical Boolean/logic circuits. These can obviously be efficiently classical simulated so they form another island. Here the relevant quantum resource that frees you from the island is what is called coherence.

There are probably a bunch more examples. I think you can invent one where all your quantum logic gates have to have at least a certain amount of noise in them, but I haven't checked the details on that.