How did physicists figure out neutrinos have mass?
Neutrinos come in three types, or “flavors”. It turns out that if neutrinos change their flavor, they must have mass: there is just no way for a change of flavor to happen if neutrinos are massless. It was in fact my own experiment, Super-Kamiokande, which made the first definitive observation of flavor change (also known as “neutrino oscillation”): we observed neutrinos from atmospheric cosmic ray collisions changing flavor, in 1998. This was the first clear evidence of non-zero mass neutrinos. Since then, there have been many confirming measurements of neutrino oscillations. Some of the above links tell some of this story.