Reply from Felix via e-mail:
Ok, so basically it is so sensitive in NA range that it picks up noise induced by the wires. There is no way around it. Remember, the output will be amplified 100 times. The leads will act as antennas. In fact if you try this on the uCurrent, the effect should be even worse (looking on the scope the noise fills the screen).
The ADC sampling is a simple averaging algorithm using built in averaging of the SAMD21.
The only way to filter out noise would be to do more advanced software filtering. In other words, to simulate the RMS algorithm of the DMM, which in fact is much much slower in terms of bandwidth, and may only give you a few readings a second.
The default firmware algorithm for the ADC, as given in the sample sketch, only samples a few times a second as well. This is to keep noise coupled into the amplifiers output at a minimum - that would only be visible on a scope though, not on a RMS-ing DMM. So if you dont care about using with a scope, than you can increase averaging, or even sample+average constantly, between OLED updates. So that should help dramatically with the jumping. However you can not expect the 12bit ADC will be as accurate as the advanced DMM ADC circuitry. It is just there to give you "good enough" informational readings. And of course, with the limitations I have discussed above - it's possible to make it more accurate/stable, but with the tradeoffs I mentioned.
I hope this makes sense?