Reversed First Joule's Law : heating a resistor produce voltage?

Nice experiment!

But think: why would a current flow in a particular direction and not the other? Is the system somehow asymmetric? As the other answers have suggested, this is probably a thermo-electric current due to the Seebeck effect (a complication I didn't want to get into for your other question).

But now that we're there, here's another thing to try: can you change which direction the current flows by how you are heating the resistor? Does the magnitude or direction of current change when you heat one wire of the resistor versus the other? The Seebeck effect comes into play at electrical interfaces, where there's asymmetry, so the direction of the interface would determine the direction of the current!

The asymmetry issue is another way how you can see that you can't reverse the standard Joule heating: You will get the same heating for a current running either direction through the resistor. But if you cool the resistor to try to undo it, which way would the resulting current flow?

I believe it may have been a manifestation of the Seebeck effect.

As some dissimilar metal to metal junction was heated in your setup, it started to generate a small voltage difference somewhat proportional to the temperature difference. Essentially it was probably a really poor and non-linear thermocouple.

Not really the reverse of the Joule Heating though.

A lot of the other answers are describing what you saw. I will describe what you were looking for. You noticed that in joule heating, the energy from moving charges gets converted to thermal energy. You can ask if the thermal motion of atoms in the resistor can create a voltage. The answer as you guessed is yes. However, since the thermal motion of the atoms is disorganized, you will not get a steady voltage pointing in one direction along the resistor. Instead, you will get a small fluctuating voltage across the ends of the resistor. The variance of the distribution of voltages you will get is proportional to the temperature, so heating the resistor up will indeed increase the RMS voltage you observe. This effect is called Johnson noise.

Since this noise is weak, you will need an amplifier to measure it. Here is a lab manual for a Johnson noise experiment from MIT I found online if you want to try to do it yourself or see how it would be done.