How to judge a potential graduate student based on a short interview?

There are some excellent answers here. I'll add my approach.

I select a research paper that is relevant to the project/area I'm looking a PhD candidate for. Relatively entry level difficulty. I give the candidates a week to read the paper and during the interview I ask a few questions:

• What problem is discussed in the paper?
• Why is it important?
• What solution did they propose and why was it better than the past ones?

It reveals to me a lot of things about the way they think, knowledge in the area, critical abilities, etc.

I think this problem is more or less insoluble. But here are some things I look out for:

• I always ask candidates to give a 5 minute presentation about a research project they have done. While there is a risk that this bias' against students who have had fewer research opportunities, all accredited degrees in my field should include a capstone project with at least some research component. The key here is that you are not looking at the quality of the results, but the quality of the presentation - in particular, does the candidate understand what they were doing and why (or why the awful project their supervisor made them do made no sense :)). This weeds out about a third of student who either don't or can't communicate an understanding of what they did and why.
• I ask them to describe a recent paper they read. You'll be surprised how many can't bring a paper to mind.
• I ask them why me - this is telling because you get a feel for whether they have read your work and understood it - that is, are they actaully interested in working with you, or did they just send their CV to everyone and anyone.

Rather than allowing you to select the best student, these hopefully allow you to weed out the worst, which I think is probably the best you can do.

Disclaimer: the following answer assumes that the prospect student will be working in a "collaborative" environment.

As stated in the comments, having the certainty that the chosen student will be the right one is impossible.

That being said, I think that the main purpose of these live interviews is not to assess the quality (in terms of skills, knowledge or expertise) of the potential candidates: evaluating the proficiency of the candidate is very difficult in such a short time; you should trust the CV and the reference letters.

On the contrary, your goal should be to understand if the prospect student is a good choice from a personal point of view. "Will I want to work with this person?", "Will this person be capable of blending in with the other people in my lab?", "Will the other people in the department be able to collaborate (if necessary) with this person?": it is questions such as these that you should try to answer with the live interview.

In order to do this, try to ask unusual questions that may induce the candidate to reveal his true self. You could even try putting some pressure on him (try asking some specific questions that you know in advance he will not be able to respond) and see how he reacts.

In summary, assuming a 20minutes interview, I would structure it like this:

• First 5 minutes: I will discuss the background of the candidate. The objective is to get the general picture of the student, and put them at ease.
• 5 to 15 minutes: I'll try to get to know their real persona. The candidate should be "warmed up", and this is the perfect time to evaluate their personal behavior and determine if they are fit or not for the role.
• Last 5 minutes: I will ask for their career goals/research objectives and similar long-term plans. This is just to conclude the meeting in a more relaxed way for both parties.

By following a similar structure, the candidate should expect the first and last steps, but the middle (and most important) step will be unknown to him.

As for what questions to ask in the second phase -- they should be domain-specific questions which either fall directly into your area of expertise (and for which very few people know the right answer), or more open questions for which no real answer exists (yet). You can start with an "easy" question, and then elaborate a discussion that will end with one of the two options I provided. Remember: the goal here is not to determine the student's preparation, but gauge their ability to deal with unforeseen circumstances.

Finally, these answers in another site may be useful to you:

• Is stress interview a red flag when deciding whether to accept the offer?
• Got into a (technical) argument with my interviewer - should I apologize?
• As an interviewee, how to handle situations where the interviewer is unprepared or asking the wrong questions?

When interviewing candidates for industry jobs, I concentrated on two issues:

1. Is the resume accurate, or if inflated how much? A resume is a sales document, and candidates do try to present themselves as favorably as possible, but it should be fundamentally truthful.

2. Can the candidate discuss technical issues in the relevant field?

Everything else is covered by the paperwork.

Fortunately, there is one form of question that can help resolve both issues. Pick a topic, such a research project they claim to have done or a course they did well on, and ask about it. Do they know as much about the topic as the resume indicates, and can they discuss it?

As Buffy suggests in a comment, zero error rate may not be possible. Ultimately, you just have to do your best and accept you will make mistakes.