Neutrinos were originally hypothesized in order to solve a problem which did not require them to have mass, and for a long time after they were actually observed, their measured masses remained within error bars straddling zero. It therefore made perfect sense to model them as massless.
But to actually include neutrino masses in the Standard Model is trivial, and was done long ago.
The most straightforward way to do it is to give them quark-like mass terms. This requires introducing a right-handed partner for each known (left-handed) neutrino, which some people don't like because right-handed particles don't partake in weak interactions, and weak interactions are the only (known) neutrino interactions (apart from gravity), so you end up with undetectable particles.
The main alternative is to use Majorana mass terms, making neutrinos their own anti-particles, which some people don't like because it deviates from the pattern of all other fermions in the Standard Model.
A third way is to say "it's both", typically involving the seesaw mechanism, which some people don't like because it requires unfashionable GUT-style beyond-Standard Model physics.
Point is, there is neither a failed "prediction" nor a great "mystery" here. There is uncertainty about which kind of mass term we should use for neutrinos, because the experimentally observable differences between the alternatives are really, really tiny.