A Fair Question...
...and one physicists and cosmologists will be thinking and arguing about for awhile. In a nutshell, the areas of science likely to be affected by this finding are:
While the Standard Model, which very accurately describes the interactions of elementary particles will most definitely not have to be rewritten, there will be less dramatic effects. The question of how particles acquire mass is one of the deepest unsolved mysteries of elementary particle physics. Neutrinos had been thought to be the only fundamental constituent of nature which did not have a mass. In light of this discovery, that long-standing belief will have to be revised. However, the Standard Model itself does not "predict" one way or another whether neutrinos have mass - this one of the many parameters of the model which must be input by hand. This is in fact one of the universally recognized shortcomings of the Standard Model, and why most physicists doubt it is the complete, final theory. A truly complete theory would predict the masses of the elementary particles rather than requiring them as inputs.
The effects of the very small neutrino mass implied by the Super-Kamiokande result will probably be minimal in terms of affecting the quantitive predictions of the Standard Model. More promising is the prospect that knowledge of the existence of neutrino masses, and an estimate of their magnitude, will shed light on the larger question of how the particles have the mass that they do. With the discovery of neutrino mass, it now appears that mass is a property common to all matter - in itself a highly significant discovery.
The problem of missing mass or dark matter has received widespread exposure. In observational astronomy, gravitational influences are evident, within and among galaxies, which exceed those expected from the visible matter (i.e. stars). Neutrinos have been suggested as one source of this gravitation, but the small neutrino masses implied by the Super-Kamiokande result may be insufficient to account for all, or even most, of it. More likely is renewed theoretical attention to the cosmological effects of neutrinos with mass. At the very least, the neutrino is the first serious particle physics dark matter candidate actually known to exist.
It has also been suggested that neutrinos played a role in catalyzing the primordial density fluctuations which eventually grew into galaxies. This question is also likely to receive renewed attention, since neutrino mass - a prerequisite for any influence of this kind, has moved out of the realm of speculation and is headed into the domain of fact.
In addition to theoretical pondering, a new wave of neutrino experiments - all using new techniques to uncover the secrets of the neutrino, is soon coming on line, and more are planned. Find out more with the link above.