While SnowTech was an engineering school to the core, my high school education was very much like attending a liberal arts college. I was enrolled in the International Baccalaureate program, which means that, among other things, I can spell baccalaureate correctly on the first try. It also means that I got a much broader education than the typical American high school student.
As a graduate student, it is pretty much inevitable that you will work with people from other countries, and absolutely inevitable that you will work with people from very different cultural backgrounds than yours. Sure, you'll have common scientific backgrounds, but there's more to an effective working relationship. This is where I think it's important to have that broader background to draw on.
My high school experience involved a very extensive study of modern (i.e., 1760-ish onward) European History, as well as a brief study of Asian and Latin American History. Learning history gives me perspective on more cultural background. More importantly, though, history class is where I really learned how to write a coherent argument. We wrote essay. Lots of essay. Some of them at home, some of them where we were given specific resources to analyze, and others where we had to recall the supporting information without help. We were taught how to structure an argument, and how to support a thesis statement effectively. Over and over, I read about how scientists can't write or communicate. Perhaps it's because they were never really taught how?
I also had to learn a second language (I chose Spanish). More than anything, I think this is something everyone should have to try, just so they can appreciate how difficult it really is. Yes, your Chinese TA may have a thick accent and trouble with articles like "the", but your Chinese would probably be even more unintelligible. Trying to speak a second language helps you learn how to effectively manipulate a restricted vocabulary. Depending on who I'm talking to, I tend to adjust my speech patterns (particularly my enunciation), because I'm aware that my standard mode of speech can be excessively florid, slightly mumbled, and non-linear. Learning Spanish made me a more effective communicator, even though I'm still speaking in English.
Speaking of English, my freshman year focused on learning how to extract meaning from text efficiently. We were assigned 26 books over the course of the year, and while my freakish speed-reading self completely read all but one of them (I just can't get into Dickens), we were taught strategies on how to look for meaning without poring over every single word. I also learned how to edit myself brutally. We were given red pens and our own papers, and she expected them to be bleeding when we turned them back in.
I've already talked about how music lead me to materials science, and it should be obvious that chemistry and physics and math were very important to that as well. However, I think the most important skills I learned in high school weren't the facts or formulas I memorized, but the communication skills I learned outside of STEM classes.