This is another question asked to gauge your values, your ability to deal with unusual situations (a strange question,) and your creativity. This question has been asked enough that now people prepare for it, so it’s not as powerful a question as it once was.
Still, interviewers ask it, so be ready for it.
It’s best to stick to people with positive images, like Dr. Stephen Hawking or Bill Gates, while avoiding clearly negative or evil people like Hitler or Charles Manson.
Avoid polarizing figures, such as political figures like a sitting President, unless you’re sure of the interviewer’s political leanings. Even popular figures like the Pope could create conflict. In general stick to the rules of polite conversation – avoid politics and religion. Even if you do, anyone might have a problem with anyone, and the follow up question could include some perceived negative such as:
“You’d want to be stranded with American Idol Simon and have him criticize everything you do?”
In those cases, if you’ve selected a seemingly neutral person, laugh it off. “Well, we’d have to set some ground rules first, no criticism, or singing.”
Name well known, rather than obscure people. The rule of thumb is if you have to explain who someone is, it isn’t as good an answer because it’s likely meaningless to the interviewer.
“James Watts, because without him nobody would be able to easily tie their shoes.” Most interviewers won’t know that Watts invented the Aglet, the now hard plastic end on a shoe lace.
The answer is OK if you throw in a brief biography. “James Watts, he invented the hard end on a shoe lace, the Aglet. He was really good at coming up with simple solutions to every day problems, which I think would be valuable on an island.”
You can mention people who aren’t famous, but who fall into positive stereotypes such as a brother, father, or other important relation or mentor.
“I’d love to spend time on the island with my former boss. He was always calm in the worst situations, I think that would be important.
Or, take the perfectly pragmatic approach.
“I don’t know if you’ve seen the show Survivorman, but this guy is an expert on surviving in the wilderness. He knows what to eat, what not to eat, how to make shelters, how to gather water, all the important survival skills.”
One answer, given by a recently passed up candidate, is a sure loser. “Not you, we’ve only been talking 5 minutes and already I can’t stand you.”