What are predictors of the success of PhD students? Every year we organize a competitive international call for PhD students (in the area of biology). What measurable criteria should we use to predict their academic success and award them research fellowships?

I realize that part of the question is ill-defined because it is not clear how to define “success” for a PhD student. But since I imagine that many of us have this problem and have potentially thought of a solution, I would love to read your thoughts on this question.
The major quantifiable predictor of success in research is... success in research. People who have done research successfully in the past are more likely than not to continue to do so. 

For students that have not done research in the past, the best predictor I have seen for success (whatever that may be) is expressed in a quote from The Unwritten Rules of PhD Research, by Marian Petre &amp; Gordon Rugg:


  A willingness to learn for themselves and good judgement about when to stop and ask for feedback.*


It's not exactly quantifiable, but you can get a sense for it in an interview. 
(Of course, this quality can be learned, so a lack of it doesn't necessarily predict an inability to succeed at research.)

* I took this quote completely out of context; the authors there are actually discussing the role of the PhD advisor, and they mention this quality in reference to a student "who can be pretty much left to get on with it, with supervisory meetings being something that both parties enjoy, and where each party learns from the other."