LVDT Measurement question
Is there any reason one might want to measure both the primary and
  secondary coils on an LVDT?


To get better accuracy is the usual reason. The amplitude is somewhat subject to noise perturbations so, if you also measure the phase angle between primary and secondaries you get a more reliable result. To do that you need to know what the primary wave form is. This is normally done by gating each secondary signal with the primary signal (converted to a square wave) and this produces a superior signal to noise ratio than just measuring amplitude directly. In effect the receiver uses synchronous demodulation to obtain the two amplitudes.