Someone once told me that if you're a grad student studying under a Nobel laureate, you're much more likely to later win the Nobel yourself. (I just searched the internet for evidence regarding this claim and couldn't find any, so I'm now less confident in it.)

Challenge accepted. Begun 8:40PM, finished 8:47PM. Damn, I'm good. Citation:

In a study conducted with 92 American winners of the Nobel Prize, Zuckerman (1977) discovered that 48 of them had worked as graduate students or assistants with professors who were themselves Nobel Prize award-winners. As pointed out by Zuckerman (1977), the fact that 11 Nobel prizewinners have had the great physicist Rutherford as a mentor is an example of just how significant a good mentor can be during one’s studies and training. It then appears that most eminent scientists did have people to stimulate them during their childhood and mentor(s) during their studies. But, what exactly is the nature of these people’s contribution.

• Zuckerman, H. (1977). Scientific Elite: Nobel Laureates in the United States. New York: Free Press.

GS lists >900 citations of this book, so there may well be additional or followup studies covering the 40 years since. Or, also relevant is "Zuckerman, H. (1983). The scientific elite: Nobel laureates’ mutual influences. In R. S. Albert (Ed.), Genius and eminence (pp. 241-252). New York: Pergamon Press", and "Zuckerman H. "Sociology of Nobel Prizes", Scientific American 217 (5): 25& 1967."

How did I find it? A few wasted searches like 'factor predicting Nobel prize' or 'Nobel prize graduate student' in Google Scholar, until I search for 'nobel laureate "graduate student"'; the second hit was a citation, which is a little unusual for Google Scholar and meant it was important, and it had the critical word mutual in it - simultaneous partners in Nobel work is somewhat rare, but temporally separated teams don't work for prizes, and I suspected that it was exactly what I was looking for. Googling the title, I soon found a PDF like http://www.pages.usherbrooke.ca/rviau/articles/principales_communication/eminent_scientists_demotivation_in_school.pdf which confirmed it (and is interesting in its own right as a contribution to the Conscientious vs IQ question).

no Nobel laureate has bothered to sit down and write a book called "How I Do Research". (Please leave a comment if you know of a book like this!)

Arguably, Feynman's various books and collections constitute such a thing. Alternately, Turing Award winner Richard Hamming wrote The Art of Doing Science and Engineering: Learning to Learn, which is just that. (I was disappointed and thought his talk was much better.)