In order to teach students what "statistical modeling" is, it would be helpful to show them some very concrete, specific (and maybe classic) examples of statistical models. Something you can point to and say, this is a statistical model. Do you recommend any examples of statistical models to use for this purpose?
Oil decline curves and EUR calculation. Lot of good articles on the web and Youtube videos. [I would link, but seriously there are so many explanations...if one is too technical or long, skip around and find another.] Very accessible (to the teacher, I would hope..little translation perhaps helpful for the students. You have a time series that is imperfect (even for a single instance) and then you have multiple instances (natural resources have variability even within a specific area). Typical regression is an exponential decline but some wells (especially shale) have hyperbolic decline (so the regression is not only getting an initial decay constant but the "b" exponent of the hyperbolic function).
Election predictions (see 538 blogsite or Andrew Gelman's blog).
Many chemical or other manufacturing processes. Sorry, I don't have an exact one to share but again basic Google or Youtube search should come up with many real data examples. This is sort of a classic type of model since you can regress temp, pressure, reactants, flow, etc. and there can be some complexity (nonlinearity, multiplicative factors). Typically rate of production of one product is the key output variable, to optimize, but it can be more complex...optimizing profit for instance, which will depend both on output of product and cost of the inducing conditions. Often the situation is so complex that there is no theoretical equation, even the machinery dimensions can change things, thus you gather data and make a model statistically rather than using a function.
Of course crop yields (this is where statistics started).
Safety models (nuclear reactors, aircraft, spaceships, etc.)
Lots of business problems ($ output as a function of salesperson skill, customer classification, etc.)
Hiring efficacy and performance prediction (Journal of Applied Psychology is the best journal for thoughtful HR statistics).
Biostatistics: both disease detection and causation.
The whole field of sports statistics. An example I like is outcome of NFL players by draft position. An analog output variable might be years in the league or a digital one is specific outcomes (made roster, started, made Pro Bowl, All Pro, etc.) The cool thing about building this model is that you actually have a lot more input gradation than just "round" (a common sports misconception) if you use numerical position. End of the first is more like beginning of the second than beginning of the first. Also, NFL has implemented compensatory picks so the picks per round is variable. A confounding variable to watch out for is size of the league (if you go back far enough).
Most of the field of climatology AND meteorology. A nice specific one is annual tree ring size as a function of temp and moisture. Lots of variability for uncontrolled factors (insects, fire, spacing, etc.) Even tree ring size on a specific tree (at a specific height) has variability as the rings are not perfect circles (usually 3 cores are taken and averaged). Cutting a section gives better control of course (and more accurate measurement, less human factor confounding...but it kills the tree!)
Student performance (ed research) as output based on specific training methods used (and population controls: SES, IQ, age, sex, yada yada).
Advertising reaction (clickthrough, purchase decision, etc.) as a function of type and frequency of touch as well as demographic customer factors: age, income, etc.
Running rats through mazes.
(Probably a gazillion others I'm not thinking of)
Just saw that you want a specific recommendation to use: I would use the chemical process. It is very intuitive and people have kitchen cooking experience to know that temp, time, etc. will affect output. This is also the first model that Box, Hunter, Hunter use as a classic explanation. Think this is an example, where you don't really need to get too much into discussing physics or industry specifics (reactants A, B, C, etc.). Think it is easier for the first example to be rather broad and hypothetical rather than using real data or a real business problem.
An alternate would be biostatistics (say cancer as a function of variables like age, smoking, etc.). This is one that is also very intuitive to students. If you use that I would keep it simple in terminology at the beginning of a course (avoid having to teach specific biology).
P.s. I don't mean this aggressively but am puzzled how a teacher would not already know some of these (or many others). Any textbook will be full of them. Box, Hunter, Hunter Statistics for Experimenters is an excellent reference. Very easy to read and full of examples.
P.s.s. Stats rocks. Remember what Hamming said ("I won't fly in an aircraft that depends on the definition of the Lebesgue Integral.") But he sure flew in some that depended on statistics!