I shadowed in neurosurgery for a few months this year and wondered the same thing. Will you be going to rounds in the morning before hand? If so (and I guess either way) I would go business casual. I always wore slacks, slip on dress shoes, and a button down (no tie or anything like that). I would always sit in on rounds then follow the chief resident through the morning and early afternoon, so right after rounds he'd check me out some scrubs to wear. I sort of wish I'd brought a pair of sneakers but you may have a lot of the day when you aren't in the OR and it's sort of awkward to carry around an extra pair of shoes.
And I'll second what the poster said above me- befriend the nurses and surgical techs! They all came to like me and would help keep me from getting lost and give me a place to sit or stand.
If this is someone you want to get a LOR from later be sure to build a good relationship with them. They won't expect you to know much but you can really impress people by asking insightful questions and generally showing that you really want to understand what you just saw or what they will be doing. Also, if this will be ongoing, just try to figure out the vibe among the docs. Everyone I was shadowing around were real ball busters and seemed to like that I would crack jokes with them.
Sorry, I know that's WAY beyond "what should I wear" but I hope it helps some! Have fun and above all just try to imagine if you could do this day in and day out. It can really get you fired up for the long road ahead!!
