Noether’s Theorem is something I wish was introduced a lot earlier than it ever seems to be: I think a simple, restricted, but still enlightening form of it can be introduced as soon as students have a basic mastery of derivatives and the idea that a derivative of zero means the function is constant with respect to that variable. You don't need to introduce the idea of generalized coordinates to give a flavor of how powerful this notion is.
IIRC when I studied physics I think they tried to tell us that there was something profound about the fact that time-invariant systems conserve kinetic energy and space-invariant systems conserve momentum, … during the first year, but I don't think anyone was mindblown. Then during the second year we started learning about generalized coordinates and Lagrangians and I think we did learn the Noether theorem then (probably just about one semester later).
However I think I really fell in love with it when I understood gauge invariance.
In general I think you can always try to tell your students things a little early to impress them, but it rarely works until they can work out the math themselves. I remember our professor in introductory quantum mechanics saying something along the lines of: "and now you see why quantum mechanics is just Markov chains in imaginary time", but until another professor showed us the Wick rotation in the context of path integrals nobody really appreciated that even if it could have been in our grasp earlier.
I guess this has come out way denser than I meant it to be, but my point is that you can always try to introduce something a little bit earlier, but you'll often find that your students don't want to learn some diluted crap, they want the real deal!
I didn't learn to appreciate Noether's theorem until my second semester in graduate school. I learned the math in undergrad and knew if you did fancy derivatives, you could obtain a conserved current. I thought it was just a way to get the conservation laws for cheap (kind of like how you got the equations of motion from Lagrangians for cheap, as was my understanding at that point).
Then, I learned about trying local U(1) symmetry and trying to make the Langrangian invariant under that and wham, you get EM for free, without even postulating it a priori. I was amazed--all of a sudden, I realized the reason why all these particle physicists were so bent on postulating this or that symmetry, hitherto not seen, might exist in nature; I realized how fundamental symmetries were and how deeply intertwined with conservation laws they were.
Though I didn't find Scott Aaronson to be as clear as the professor who finally showed us the Wick rotation. Unfortunately he passed away a few years back right after having missed being awarded the Nobel prize by an inch.
Eugene Hecht introduces it conceptually at the beginning of his algebra-based College Physics book. It's nice to capture the attention of the women in the class with this person (woman) of enormous import that "nobody ever heard of." Bravo professor Hecht!
Here's a few simple examples: http://www.sjsu.edu/faculty/watkins/noetherth.htm
Nothing about this requires math beyond undergrad calculus.