There are several motivations to search for possible non-zero neutrino masses [1].
Fermion masses in general are one of the major mysteries/problems of the standard model. Observation or nonobservation of the (oddball) neutrino masses could introduce a useful new perspective on the subject.
Nonzero 
masses are predicted in most extensions of the standard
model. They therefore constitute a powerful probe of new physics.
There may be a hot dark matter component to the universe. If so, neutrinos would be (one of) the most important things in the universe.
The observed spectral distortion and deficit of solar neutrinos is most easily accounted for by the oscillations/conversions of a massive neutrino.
The ratio of atmospheric 
may be suggestive
of neutrino oscillations.
There are hints for possible 
oscillations from the LSND experiment at LAMPF.
With or without neutrino mass and oscillations, the solar neutrino flux is (with helioseismology) one of the two known probes of the solar core. A similar statement applies to Type-II supernovae.
Although there are strong motivations for neutrino mass and mixing from
theory, cosmology, and astrophysics, the number of types of neutrinos is
limited. The LEP lineshape measurements imply that there are only three
ordinary light neutrinos, and big bang nucleosynthesis
severely constrains
the parameters of possible sterile neutrinos (which interact and are
produced only by mixing). There are only a limited range of
well-motivated possibilities for neutrino masses and mixings. The new
generations of laboratory and solar neutrino experiments should be able
to cover this range and either clearly establish non-zero masses (probably
the first break with the standard model) or else falsify the interesting
possibilities.