So, here's the sentence:
They usually ask: if you were stranded on an island, what’s the one thing you’d wish you had?
So is it have or had in this example, and why? (if it even makes a difference)
A finite clause complementing wish is almost always hypothetical (irrealis, 'subjunctive'):
He wishes (that) he had a million dollars. but not
He wishes (that) he *has a million dollars.
The irrealis quality is often expressed with a past-form modal verb
He wishes that he might/could/would win the race.
The only exception is situations where wish in some sense "effects" the desired outcome. In a fairy tale, for instance, where the genie grants the hero a wish, he might say
I wish that I may marry the princess . . .
and the story proceeds inevitably to the nuptial.