A very interesting idea!
Although I don't think that the black hole cared much of those few photons from earth.
Expanding it, how long until my hand waving could cause a strong enough "butterfly effect" for an observable cosmic phenomenon to occur beyond our solar system?
A million years? 100 million?
Edit: Thank you for your comment. Just keep it! Sometimes people here are just a bit quick to judge. It's amazing to think humans could have affected such a monster at all, no matter how little. Even for a handful of photons worth.
My rough estimate says around 9 years, and 3 months.
The essence of chaos is that effects grow exponentially with time. For example with weather, errors in measurement lead to roughly an order of magnitude in errors in measurement every 3 days. Going from the scale of a single atom to the scale of a star is about a range of 10^60. At that rate of exponential growth means that in 180 days, which is 0.5 years a single atom out of place leads to a different outcome for things like solar flares.
Therefore a photon from your hand moves one atom on Alpha Centauri about 4.367 years later. A half-year later it causes the difference between a solar flare being there or not. Then 4.367 years later it comes back here. And the result is that in roughly 9 years and 3 months, one photon from your hand could have caused a visible change in another star.