Does the university name really matter?

It depends on the field you are going into and if you are continuing your education. Somebody having heard of your school and knowing good things about it is always a good thing, but not knowing about it isn't always a bad thing. There is at least some truth to what your friend said, but it would be better to get great marks at the little school than to not do so well at the big school because you don't like it there,