I'm the source. The calculation is not particularly difficult although it requires some knowledge of basic physics. The number is of an "order of magnitude" accuracy because it involves recasting photometric quantities into energy units and assumes some values for a "typical" film. Annoyingly, I can't find a reference to anyone else having done the sums. I'd appreciate it if you or someone else on APUG would run the numbers and check my answer.
Haha sounds like something I'd inflict on my poor students!
First of all, let's be clear: photons don't have mass so the argument that they deposit mass would be incorrect.
However, photons
do carry a momentum, namely p=E/c, and so you could pick ~532 nm as your center wavelength and that is, what, 2.3 eV energy, roughly.
So then the argument would be that the impulse F*t equals the change in momentum imparted by the photons. Each photon contributes a momentum of roughly 2.3 eV/c or 4x10^(-19) kg m^2/s^2 divided by 3x10^8 m/s, so let's say ~10^(-27) kg m/s . Then in one second you get an effective force of, what, 10^(-27) Newtons. You set that "weight" equal to m*g and use g= 10 m/s^2 to get a mass of something like 10^(-26) kg. Then you say you get a thousand photons deposited on your film in a second, and you arrive at 10^(-23) kg.