I would say to be specific about what you're interested in... or honest if you're interested in doing anything! As a medical student with some prior knowledge, you can choose to be a bit pickier than an undergrad. However, I wouldn't demand things like starting your own project or being published or whatnot... at least in my case, I was just curious if I could get started in any way. For one example, this is one e-mail that I sent (redacted a bit to preserve the small shred of anonymity I have left on this site).
Dear Dr. X,
I am a current first-year medical student at __. I have an interest in conducting research during my time here, and thoroughly enjoyed your lectures this morning on ___! I also noted that your lab is listed as one that may take pre-clerkship medical students on for research projects (possibly via the ___ research program).
To tell you a little about my research background:
I graduated in __ from __ with a degree in biopsychology. Before beginning my time at __, I was a laboratory manager in the __ at __ (P.I.: __) where we studied memory, aging, and emotion with cognitive neuroscience methods. I worked with ___ for three years, and published several neuroimaging papers.
I am currently CITI certified in biomedical research, was trained in MRI operation (functional, structural, and DTI imaging sequences) at the ____, and have practical experience and training in TMS and EEG/ERP methodology. I was also a volunteer in the ___ Hospital emergency imaging department, where I learned about clinical and diagnostic imaging.
Though I have a background that is pretty focused on cognitive neuroscience, I thought that your lab would be a great place to make a transition towards more clinically relevant work. I am particularly interested in any work with clinical populations, and, most specifically, imaging work. Please let me know if you may have any opportunities for research in the upcoming semesters -- or if you have any questions for me.
I have attached a copy of my CV for your reference. Thank you for your time!
All the best,
Johnamo