If Neutrinos really do have mass, could the combined mass of all the Neutrinos present within an entire Galaxy account for the extra mass required to provide enough gravity for a Galaxy to keep its shape?

In princinple, that idea has been investigated in the past. Problem is, that a combination of observations and computer simulations suggest that a large share of dark matter must, on a per-unit basis, have large mass. That is generally considered as ruling out Neutrinos as dark matter. For the Google alternative Wikipedia, "hot dark matter" and "cold dark matter" may be suitful terms to start looking up more details.

This was a big idea at one time. I think, as timo says, neutrinos are now thought not to be the major component of dark matter.