hi,
A LM317L is not good enough as a voltage reference for mVolts, it only a voltage regulator.
If the input to the LM317L falls due to the Relay drawing current from the 12V line, the output of the LM317 will change by a mV or so.
Look at the TL431 or equivalent to provide a stable reference.
The problem could also be caused by the way you have connected the 0V lines to the different parts of the circuit.
example. If the LM317 0V is connected so that the relay current flows via the LM317L 0V line, every time the relay is energised the LM317L 0V will be raise by a few mV,
EDIT:
This image shows the effect upon the mV source when the 0V lines are incorrectly terminated