Since the Earth rotates around its own axis, you're oscillating closer to and further from the Moon every 24 hours. When you're "facing" the Moon, it's puling you up (from the Earth). When you're away from the moon (on the opposite end of the Earth) it's pulling you down (toward the Earth).

This (along with other factors) causes tides. And tides are already used to generate power.[1] Including using the piezo-electric effect (not sure how with what efficiency, though).

[1] https://en.wikipedia.org/wiki/Tidal_power

You could carve a few cubic kilometers of rock. Raise it when the moon is at zenith. Lower it when the moon is on the other side of earth.

There was this idea to use a carved rock cylinder for pumped water energy storage (have the water under the cylinder).

You could even combine these.

Of course, in reality the difference in gravity is so tiny, the round trip efficiency is too low, killing the idea.