[Heather Jameson (OP)]

Paloma and her son are having a debate:

My son asked: "how far can a bullet travel until it stops?"
We then discussed if it would be farther in hot or cold air.
I said cold, he said hot.

Who do you think is right and why?

[Colin2B]

Hot air is less dense so a bullet will travel further.
However, the air is less stable, particularly near the ground, so accuracy suffers and although the bullet will go further you have a lower chance of hitting the target.
So you need to reword the question and you can be right - if that matters 

[evan_au]

how far can a bullet travel until it stops?

If you fire a bullet in outer space (beyond Earth's atmosphere), it can travel for millions of miles without slowing down, let alone stopping.

Of course the concept of velocity is a very relative thing when you are in outer space, as there is no "obvious" frame of reference from which to measure it.

[alancalverd]

At constant pressure, the viscosity of gases increases with temperature, so the bullet will travel further in cold air.   

[Bored chemist]

In principle, in space you can fire the bullet into orbit in which case the distance is huge (not quite infinite- there are energy losses due to gravity waves, and tides etc. The question of which effect dominates is left to the interested reader)

[Bored chemist]

This is correct

At constant pressure, the viscosity of gases increases with temperature, so the bullet will travel further in cold air.   

(Actually, the pressure has relatively little effect on viscosity)
So is this

Hot air is less dense so a bullet will travel further.

And I genuinely don't know which effect will win.

[Colin2B]

This is correct

At constant pressure, the viscosity of gases increases with temperature, so the bullet will travel further in cold air.   

(Actually, the pressure has relatively little effect on viscosity)
So is this

Hot air is less dense so a bullet will travel further.

And I genuinely don't know which effect will win.

Ah, the plot thickens ...... or is it the air?

I do know from experience that (at same altitude) bullet will drop below point of aim if cold and above if hot. Ballistics calculators put that same compensation in.