Very small noise when measuring small current with ADS1115

This is of course an Adafruit or Sparkfun board you have.
Not one from ebay or Amazon, with most likely a rejected or fake chip.

The ADS1115 is an absolute (voltage A/D.
I see you want to measure low currents.
Shouldn't you look at something like the uCurrent Gold from EEVblog.
Leo..