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.
I'm not sure exactly what OP is implying though.
 light, radiation, information, etc.
 ie, reflected, or caused some reaction
The destruction of Atlantis?
You're looking at something more like 50,000 years ago, so something like the Toba population bottleneck (which probably didn't happen 70,000 years ago).
The cleansing of saidin in the previous turning of the Wheel, might work.