Here's one way to answer that question:
The Richardson / Einstein-de Haas effect.
The idea is simple: An electron is like a little
bar magnet. Indeed, the magnetism of an ordinary
macroscopic ferromagnet comes from a whole bunch
of electrons lined up inside it, all spinning the
same way.
If you change the magnetization, you change the
amount of spin, and that's enough to make the whole
bar magnet rotate. It's not a super-huge effect,
but it is readily observable with a simple torsion
balance. The apparatus is almost cartoonishly
simple:
www.unm.edu/~physics/demo/html_demo_pages/5H5010.html
The basic version revolves around a "magnetically
soft" material, plus a coil.
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I can imagine fancier versions, more informative
but harder to explain, using "magnetically hard"
materials (i.e. with a nontrivial B versus H
hysteresis loop). This allows you to show that
the angular momentum depends on the magnetization
of the material, not on the applied magnetic field.
One version uses no coil at all, but starts with
a "hard" magnet and heats it above its Curie
temperature. Shazzam!
====
At an intermediate level of fanciness, you could
use a material that is saturable but not particularly
hysteretic. You can start out in the linear part
of the B versus H curve, and then for extra credit
explore the nonlinear part.
=================================
If that doesn't convince 'em that electrons really
are spinning, I don't know what will. Sure, it's
nonclassical, but that doesn't make it any less
real. Actually, as I see it, the real world is
quantum mechanical. The classical approximation
is what's not real; it's only an approximation to
reality.