Getting a consistent and precise voltage for sampling
If it were treated as gaussian noise certain loose estimations could be made: -


Peak to peak of the sample is about 10mV
Divide by 6 (sigma) to estimate RMS -> 1.7 mV RMS
Sampling rate is unimportant because it will fold down noise from the upper regions due to aliasing: -




The picture above has an arbitrary Y axis showing power but, the important point is the relative power of the aliased noise compared to the unaliased noise. Picture taken from this document entitled "Aliasing in 1/f noise spectra".

How much noise does the LM317 produce? The DS doesn't say and most linear regulators don't say either because they produce noise that is of no consquence for their intended target circuit.

How much noise does the ADC have in the LCP1768. I had a quick look but I didn't seem to find the figure but, inbuilt ADCs are usually quite crappy.

Altogether I think 1mV or 2mV noise isn't unreasonable given what you say you are using.

Have you tried using a low pass (anti-alias) filter such as a 10k and 10uF capacitor to see if the noise reduces?