I have a voltage signal coming from an instrumentation op-amp between -2 and +2 volts (typical voltage might be 0.153 V) with a 30 mV high speed (~10 ns or 100 MHz) noise included around the voltage level. Is there a way to either remove most of this noise or average the signal over time in an analog way? The ADC I am using is much too slow to average out this noise.

Can't you design a high frequency, low-pass filter in series/parallel to do this? Design your filter to attenuate anything twice the frequency of your ADC resolution. Since the resolution on your ADC is too slow, simply put your filter before the input.

Filter design calc:
RC Low-pass Filter Design Tool (http://sim.okawa-denshi.jp/en/CRlowkeisan.htm)