Avoiding the appearance of name dropping in fellowship applications
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.