My question is: do medical schools like to see you involved in research by being a research assistant or by doing your own project? I am an honors student so I’ll be doing a thesis. I know senior theses don’t matter a ton, but does the research for them count? Say I find a mentor to help overshadow my research and then I’m able to come up with a great project. Would this count for “research experience”?
Not an adcom, but from what I've seen adcoms say on this forum, research is important because it teaches you the scientific method. So if in your senior thesis you design an experiment, test your hypothesis, analyze the data, etc., that should count as research. You should understand the background of your experiment, what gap in knowledge your experiment is trying to fill, and why you are doing what you are doing.
I would also point out that being a paid, full time research assistant does not preclude being able to have your own project. There are research assistant jobs out there where you'd be mostly washing glassware or labeling tubes, and that would not count as research. I am a research assistant now, and while I do my share of mind-numbing work that is necessary to maintain a lab, I have also been able to lead projects, be a first author, have my name on posters, etc. If you are interested in a research assistant position after graduation, it can be a cool gig.
In my personal experience, I don't think it's that rare to get your own research project. What happened in my lab was that after I mastered a certain skill - say surgery, I would meet with my PI and he organized all his undergraduate students certain roles. I worked with him to create a research project that would be one figure in the paper (the other figures would be from other undergrad students). Ultimately, you gather the data yourself but present it to your PI. YMMV.
This is more or less true in my lab as well. We have multiyear grants, but within those there are individual aims and many potential side projects that an undergraduate could contribute to in a way that would be considered research. But they obviously have to master the skills first. Unless you write your own grant and get it funded, your "independent" project will likely be something closely related to what the lab is already doing. For example, I wouldn't advise walking into an Oncology lab and asking for the PI to mentor your project on spinal cord injury.
OP, as you go about finding a mentor, remember that what for premeds (not directing this at you) may be a couple months of work to check the research box, is the PI's career that they have put their blood, sweat, and tears into. I'd recommend approaching potential mentors with an air of "I'm really interested in researching xyz, and I noticed that your lab does some of this research as well. Is there a way that my thesis idea might fit in with your work?" and NOT "I need an independent research project for my med school application, please help." Even if you aren't interested in research and are only there to say you've done research, do your best to appear engaged and interested in what is happening around you. The PI will be impressed and this will help you in the long run.