Sound absorption and vibration damping are two different things so it depends what you are looking for. For damping of vibrations a viscous type behaviour is desired and materials like bitumen are used (materials with viscous flow rather than an elastic response). The viscous flow converts the energy to heat whereas an elastic material gives back the mechanical energy you put into it. The way to quantify it is to measure the tan delta of your material in DMTA. This is the loss factor. It will vary with frequency and temperature so you have to measure over the range of conditions you expect the part to experience.
Foams are good for absorbing sound and so are fluffy non-wovens.