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It can look dumb, but I always had this question as a kid, what physical principles would prevent this?
Nah, I prefer using quantum spookiness for that. Send a steady stream of entangled particles to the other person on the moon first. Any time you do something to the particles on Earth, the ones on the Moon are affected also. The catch is that this disentangles them, so you have only a few limited uses. This is why you want a constant stream of them being entangled.
This doesn't account for blinking.
If your friend blinks, they won't see the light, and thus would be unable to verify whether the method works or not.
But how does he know when to open his eyes? He can't keep them open forever. Say you flash the light once, and that's his signal to keep his eyes open. Okay, but how long do you wait before starting the experiment? If you do it immediately, he may not have enough time to react. If you wait too long, his eyes will dry out and he'll blink.
This is just not going to work. There are too many dependent variables.
I don't see this mentioned in any of the other comments: the repulsion between atoms that causes the movement to propagate through the stick is actually communicated via photons. So your push really generates the same kind of particles that your light torch is generating, and they travel at the same speed. Except in the stick it is slowed down by repeated absorption and excitation by the electrons of the atoms.
What about the mass of that stick? Inertial doesn't care for your little silly games.
Neither do the two gravity wells the stick spans. And the earth and moon are moving relative to each other, someone would probably get their head knocked off by that stick. Before it eventually falls to the earth with quite a bit of force because earth's gravity well will win. Then it'll eventually settle into a giant teeter totter, assuming it is rigid enough to survive the impact.
The pole would basically be a space elevator. I suspect gravity and inertia would effectively keep you from moving the stick. Even if you could move it, you'd only be able to move it at a speed that would seem like it's stationary. As such, the light would still be faster.
The issue is, that kind of stick wouldn't even exist. You'd have better luck with between some dwarf planet and its satellite, since the stick would break under its mere weight.