Avoiding the appearance of name dropping in fellowship applications When describing my research experience in fellowship applications, I will write something to the extent of "I began pursuing 'item X' under 'Professor Y'." Of course, I follow up with why I found the experience meaningful, and what my role was.

However, I have received feedback from at least two different sources saying that this name dropping is frowned upon, and that if I have a panelist who hates this, that it will torpedo my application. 

In my mind, the person I am working with may be relevant if a panelist knows of their work. However, at the same time, a fellowship is about me. Is there a good reason to include the names of the PI's, and if so how can that be written? Or is it truly safer to just remove the names of PI's, especially if they are someone famous (e.g. an NAE/NAS member)?
The term "name dropping" represents an excessive use of auxiliary authoritative figures, with no justification or substantive basis, to promote one's image.

This means that if you say that you worked closely with Figure X, and you haven't, you are name dropping. But if you did work closely with Figure X, it is not name dropping to mention him/her. That is, a name dropper is an imposter, while someone who lists relevant names with sufficient justification is not a name dropper.  

Hence, the advice you got from your colleagues regarding "a panel that hates name dropping will kill your application" is probably misguided and false.