The solar neutrino problem concerned a large discrepancy between the
flux of solar neutrinos as predicted from the Sun's luminosity and
measured directly. The discrepancy was first observed in the mid-1960s
and finally resolved around 2002.
The flux of neutrinos at Earth is several tens of billions per square
centimetre per second, mostly from the Sun's core. They are
nevertheless hard to detect, because they interact very weakly with
matter, traversing the whole Earth as light does thin air. Of the
three types (flavors) of neutrinos known in the Standard Model of
particle physics, the Sun produces only electron neutrinos. When
neutrino detectors became sensitive enough to measure the flow of
electron neutrinos from the Sun, the number detected was much lower
than predicted. In various experiments, the number deficit was between
one half and two thirds.
The problem is practically solved today. It wasn't so until $\approx$ 2002. However, a little chance that we know something not enough well, still exists, but it is little.
The currently accepted solution of the problem is this: yes, we know the Sun enough well, but the neutrinos can change flavor on the way they reach us. Thus, for example, if an electron-neutrino becomes muon-neutrino, then the detectors capable to see only electron-neutrinos, won't detect it.
It is today a hot research topic in the particle physics, thus some surprising result in the future isn't yet closed out. The main problem is that the neutrino practically don't interact with anything (a light year of lead would stop half of them), so there are only very little possibilities to measure their properties.
KATRIN is a German acronym (Karlsruhe Tritium Neutrino Experiment) for
an undertaking to measure the mass of the electron antineutrino with
sub-eV precision by examining the spectrum of electrons emitted from
the beta decay of tritium. The core of the apparatus is a 200-ton
spectrometer. In 2015, the commissioning measurements on this
spectrometer were completed, successfully verifying its basic vacuum,
transmission and background properties. The experiment began
running tests in October 2016. Regular measurements started
2018-06-11, with a projected duration of 5 years.
As far I know, too much surprising results aren't expected from this - most likely, the electron neutrino mass of some tenths of eV will be found, or possible the result will be that the experiment is not enough precise to find anything. (In this case, we will still get an "upper limit").
However, there is a little chance that some surprising result will happen.
For example, such a neutrino mass could be found, which can be explained only by some hyphotetical effect (like sterile neutrinos, which might also contribute to the solution of the dark matter problem).
Go back to the 80ties, where a possible solution to the solar neutrino problem was a different working and internal structure of the Sun, which looks the same for us, but generates lesser neutrinos. By analogy, a similarly different Sun should be expected, which goes into a red giant phase soon. (Or, at least its luminosity will increase enough in the near future to make the Earth inhabitable).
Note, we also know a lot of other stars today, there are very good models running on big computers, and these are enough sophisticated to close this possibility out with a good probability. Thus, the Sun should be somehow exceptional on these results.