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I think it's only fair that we assess this question from the other side: to deserve a good phone call or a nice letter from their PI, what does a graduate student or a postdoc need to do? Here's a small list:
Results: It's brutal, but it's true. If you're productive, things will come your way (hopefully.)
Effort/perseverance: Barring lots and lots of results, I think it's fair that you throw the kitchen sink of effort at your work. This doesn't mean chaining yourself to your hood, necessarily -- I think it means a thorough assessment of the problem you're working on, from the literature on down. You have to be able to answer the typical PI (and interview questions!) that start with... "Did you try [list of random but obvious techniques]?"
Leadership/mentorship: It could be as simple as setting a good example, or it could be as complex as training new graduate students in the lab's special techniques.
Of course, all too often, students' contributions to their groups are missed; and true, some PIs are just clueless. In those cases, of course, you're really in trouble.
Readers, I'm positive I've missed something. What does a grad student or postdoc owe their PI?
UPDATE: Liberal Arts Chemist, an actual PI, writes in the comments:
Loyalty: I would not want blind loyalty or an unsafe loyalty but in the ups and downs of the PI - research group relationship I have seen too many students piss in their own well. Departmental politics are brutal and a tired and bitter graduate student can really cause problems if they find a kind and listening ear in a competing faculty member. This is the kind of loyalty that feeds to departmental tribalism and in fact will do the grad student no good at all. It will come back to them either through faculty - faculty gossip or a less than supportive letter of reference. Students need someone in their lives to whom they can download their "Bitter" file ... but that person should not be in their department.For what it's worth, I agree with him.