Well, we ended up just doing additional tests before heading home:
First we took the clean signal (green box) and wired it to a free analog input (a test point on the PCB) to see the same noise was added, it wasn't, the signal remained clean
Next, we took an external buffer we had previously made on a perfboard, same schematic, and fed it a clean 1V signal, whose output we put into one of the free analog inputs, then tested the signal coming into the arduino with the oscilloscope, that signal remained clean as well
We're still pretty stumped as to why we are getting this noise (which doesn't look random since the frequency/amplitude is the same in each case) is added to the signal.