Damn those in-laws. Mine have to put up with it
That might actually have been the issue. If it is, phew! Good find! 1mV or so of noise is pretty good. My Thurlby TS3022 is around 2.5mV noise. I suspect that needs refurb as well but I can't be bothered at the moment
Inductance is hard to visualise and think of really. There are two domains to think of it in; the time and frequency domain. In both domains it is exactly the opposite of the capacitor. Fundamentally in the time domain, if you stick a voltage across it, the current rises from 0 to infinity exponentially at a rate defined by the inductance. When you remove the voltage source, the magnetic field collapses and generates a high voltage to try and force the current to flow again (this is incidentally basically how switching power supplies work). In the frequency domain it's a frequency dependent resistor. The higher the frequency, the higher resistance (generalised as reactance in these things). Typically outside of RF, which is purely concerned with high frequencies, they are used to get rid of high frequencies like oscillations which is why you see a ferrite bead on a transistor leg occasionally. That's a little inductor that stops high frequency oscillation.
You have three modes that could have caused noise which I will detail:
1. The loop acted as an antenna (electric field coupling). This could be a resonant length of some RF flying around in your house. This would develop a voltage across it. At that size this is possible but unlikely I would suggest. Anything resonant at 166KHz is usually pretty big as the wavelength is 1.8Km!
2. The loop acted as an inductor (magnetic field coupling). Try this as a demonstration. Take your oscilloscope's probe, stick it on 10X if it's switchable and clip the ground wire to the tip creating a loop. Set it on the lowest volts/div setting and go and wave it around laptop chargers, televisions etc and see what happens. Basically the magnetic field lines of force will cut the loop and induce some current. Because the probe impedance is so high it'll result in a visible voltage. Some EMC testing tools actually use things that do this on purpose.
3. The additional inductance of the wire created an accidental resonant oscillator somewhere with a capacitance inside the device.
Have fun and let us know if anything else goes wonky