OK, watching surgeries certainly does not have to be boring!
This was something that a lot of my fellow interns at the surgical shadowing program said, and I never understood it until I found out that they were simply sitting in the corner watching the back of the surgeons' heads. Here are some ways to avoid that:
1. SHOW INTEREST. Perhaps it was because I was at an academic hospital for much of my OR shadowing, but generally the more interest you show, the more people are willing to spend their time talking to you about what's going on. The trick is to manage to show interest without annoying anyone (particularly the attending...they should basically not have to acknowledge your existence until they choose to on their own). Don't talk to the people who are busy...find someone, anyone, who is not busy at the moment - the med students, equipment techs, the circulating nurse, etc. - and ask them questions that show you've been paying attention. Ask them about their specific roles, how their equipment works, what they're doing with this patient that's different from other patients and why the difference?
2. Pick a good spot. This largely depends on the nurses, tbh, but if you've been friendly and polite to them from the beginning and shown them that you understand the concept of sterility, most of them (some are unbendingly overzealous), and you ask before picking a decent spot, you'll usually be OK. One thing I always did was to find something nonsterile and stand with that...it's space that was already filled, already being avoided by those who are scrubbed in, and therefore you are not adding inconvenience, and often the path in front of you is a bit clear. If you're friendly with the anesthesiologist, this is cake - they'll set you up behind the curtain. But other good options can be the tower at the foot where all the electronic/vacuum lines come in, or (and this was my personal favorite) lightboxes. If your surgeons are using headlamps, there will be a lightbox only a few paces behind them. I would stand right on its base, which let me look right over the surgeon's shoulder (at a bit of a distance), in a known non-sterile area, with nice, warm air blowing over me from the lamp (ORs are cold).
3. Help if asked, but never until asked. You don't know what's going on or how to do most of the tasks as well as anyone else in the room. However, from time to time you probably will be asked to do something simple (like get a warm blanket, or move the patient's floor bed out of the way, move a lightbox, unplug the surgeon's headlamp, etc.) Pay attention to what everyone does around you so that when they ask you for something simple like that, you can do it. This helps keep you in peoples' minds as someone who is paying attention and helpful; it makes them more likely to address you, ask about you, answer your questions, etc. down the road.
4. If all else fails, it's time to get a little devious/annoying. I stumbled upon this by accident, but it works more than you'd think. If you can't really see or find out what's going on...ask someone who doesn't know. Ask them questions that you would ask the surgeon, were they speaking to you...you'd be surprised how consistently people jump in when they hear someone giving a poor/incomplete/fumbling answer to something they know very well. Next thing you know, they're telling you about the surgery, telling you to get a better vantage point so you can see, etc.
Big hospitals are good because there are more people to talk to, but little ones can be great because you and the scrub nurse are the only people for the surgeon themselves to talk to, and they probably speak to the nurse every day. Mix it up, try different places and specialties and hospital types. You'd be surprised what you can see in the OR, if the circumstances work out. I had a group of ENT/Plastics residents, who had finished a neck dissection and were waiting for the attendings to come in to graft in some vessels, decide to quiz me on the anatomy of the neck, letting me close and pointing out different structures. For the rest of the surgery they would describe things to me (usually at least one resident was off the table since it was such a big surgery). I had a vascular attending who had people step aside for second before closing so I could step forward and see the graft they'd put in to bypass an aortic aneurysm. When I was friendly with the scrub nurses, they'd set aside gallbladders, spleens, colons, ribs, tumors, etc once they were out of the sterile field to allow me a closer look/touch. Etc., etc, etc. There is plenty to see as long as you don't give up on it for those few frustratingly dull ones (CT surg, ugh...so many people you'll never see anything or get close...just pick the brains of the pump tech. The pump is like 90% of heart surgery anyway, and it's all you'll be able to see).
And, as stated above, don't forget the rest of being a surgeon. If you shadow one, don't just stick to the OR. If they'll have you, go on rounds, go to clinic, go when they talk to the patient/family before/after the operation, etc. The interns will tell you how little the OR factors into their daily lives...believe them, and keep it in mind.