Uh, I'm not sure it's clear what you're asking. What you MEASURE in QM are probabilities; the process of obtaining these probabilities removes the phase information from the wavefunction. When you have a superposition of stationary states though, the phase of one stationary state relative to another can play a part in the time evolution. This is the basis of quantum control etc. There were some cool papers in 2002 (I think) in Nature from Phil Bucksbaum's group at Michigan where they "measured" the phase of a wavefunction through these sorts of experiments.
Phases of wave functions most certainly do not play a "wee" role in quantum mechanics. The time evolution of the wave function is a little more complicated than a phase, but it is that simple for stationary states. It also comes up in the context of magnetic fields, such as the Aharonov-Bohm effect, which is a purely quantum effect. The constraints on the phase provide a variety of requirements for systems that are closed on, say, a loop.
There's a huge amount of stuff that's contained in relative phases of wave functions, and I suggest you poke around a bit more. Sakurai talks about it a bit in the second chapter.