Filter-Amplifier circuit for Low Voltage I/P circuitry
It looks like the sensor may have insufficient signal to noise ratio over the frequency range of valid signals.  You also have a secondary problem in that the sensor signal of 1 to 2 mV is way too small for the 0 to 3 V A/D range.  That can be solved with amplification.

It looks like your signal period is a bit over 20 s.  It's not a sine, so some harmonic content looks to be important.  You haven't given response time or frequency range requirements, but let's say anything past 2 Hz is noise.

Since 2 Hz is very slow for any microcontroller, you can use a rough analog low pass filter, sample much faster than you need the data, then low pass filter in the firmware.

For example, let's say you can sample at 10 kHz.  That's 100 &micro;s per sample, which is a long time even for cheap and small micros.  Two poles of R-C low pass filtering at 500 Hz won't matter to the signal at all, but will squash signals above the aliasing limit of 5 kHz.  For example:



The two poles are actually nominally at 480 Hz with these part values.  This filter would go between the sensor and the amplifier that drives the A/D input.  It's purpose is not make the final filtered signal, but just to ensure that there is no aliasing when the signal is sampled at 10 kHz.

In the micro, you then run a low pass filter that squashes frequencies above 2 Hz.  This digital filter can be more complicated if needed, and won't suffer from analog part tolerances.  I'd start with two poles of low pass filtering at 2 Hz.

It is unclear from your graph what signal to noise ratio you will get this way, but try that and see.  If you substantially reduce frequencies past the valid signal and still don't have enough signal to noise ratio, then you need a better sensor, or you need to reduce the bandwidth requirements.