Don't think of it as name dropping. I'm sure you wouldn't say "As a friend of Dr. X, a staff psychologist at your site, I am quite familiar with technique Y and would like to learn how to be more effective in using this technique with Z."
That being said you can say, "I would be excited to work in the traumatic stress track, as this could afford me the opportunity to work under the supervision of Dr. X and learn more about Prolonged Exposure and Cognitive Processing Therapy."
My point is that if the staff psychologist is involved in the training of residents and you would be interested in what he/she has to offer, then mention this just like you would in any other cover letter. Every cover letter I wrote talked about rotations and specific members of the training faculty.