Why does my clock strip flash weird when the second hand ticks?

That flicker is almost certainly caused by electromagnetic interference or power line noise coupling into the data line. The clock mechanism, even if running on AA batteries, generates a sharp mechanical twitch or brief electrical spike when that coil clicks, and because your components are packed tight, that transient energy is hitching a ride right onto your data signal. Moving the power supply out of the case is the most definitive fix you could do. If relocating the entire power source isn't an option right now, you need to focus on physical separation and shielding. Running the data line along the outer surface of the housing, away from the bundled power and movement wires, reduces inductive coupling significantly. Even better whould be routing the data wire to a point where it can be run along a separate metal surface and then terminating it inside the clock, keeping the signal away from any high-current paths inside the movement compartment. Another approach involves using a twisted pair of wires for the data signal itself. Even if you're using a single data pin from the ESP32, twisting the data wire with another unused pair of wires inside the housing acts like a rudimentary shield, which helps cancel out the stray magnetic fields generated by the clicking mechanism. This is a common trick when running sensitive digital signals near noisy analog or mechanical devices. It won't be perfect, but it often cuts the noise enough that the blip disappears.

Trying a different GPIO pin on the ESP32 won't solve this unless the specific pin has a known antenna or is routed near a high-frequency component that's currently acting as an unintentional receiver for that noise. The problem is almost certainly physical proximity and shared power space, so isolation is the main goal. Try twisting the data wire first; it's the least invasive step.

I think stating that switching the GPIO pin won't help is a bit of an overstatement. While the wiring proximity is defintely the bigger issue, different ESP32 pins have varying electrical characteristics internally. Sometimes a specific pin is simply more susceptible to noise leakage or transient spikes, even if it isn't physically touching a power rail. I've had a couple of projects where moving the data line to another pin cut the jitter even when the physical routing was identical. It's not that the noise is caused by the pin itself, but the pin might be receiving the noise more efficiently than another option. Trying a different one is still a cheap and fast diagnostic step before you commit to running new shielded wire or moving the whole power supply.

I wonder if the transient spike is more of a mechanical shock artifact rather than pure electrical noise, which might be the case with that old stepper mechanism. Since you are using a separate USB power supply for the ESP32, I suggest adding a large electrolytic capacitor across the power input lines. This decoupling component acts as a reservoir, absorbing those brief voltage drops or spikes that occur during the mechanical click before they can propagate through the rail. This could smooth out the power delivery even if the ESP32 is technically shielded from the direct wiring interference. It is a relatively simple addition to the power input itself.

Update: fixed it. Took the third poster's advice and put an electrolytic cap across the power input, and that alone calmed most of it. Separating the wire bundle away from the clock coil killed the rest, so the noise was coming from both spots. Zero flicker now, just a clean glow every second.