In my opinion, the sentence:

I sleep in the night.

can be simply translated, as you said, by:

Je dors la nuit.

However, this turn of phrase seems a little weird to me, in France, we use this more often:

J’ai (bien) dormi cette nuit.

Yes, ‘I sleep in the night’ can and likely means ‘je dors la nuit’.

But your son might be right : I think the canonical English phrase would be ‘I sleep at night’, so maybe the English sentence has a different meaning?

To sleep in, to sleep where one is employed (or where one spends the day).

Je dors sur place la nuit.

meaning : I sleep there (sur place) at night.

Or even : to sleep in, to oversleep.

Je fais mes grasses matinées la nuit.

meaning : I sleep in at night (in the case of someone working weird shifts I guess.)

It is a context thing, pick whatever you feel works best in your situation.

Je dors la nuit is correct French but a small change can turn it to a more idiomatic way of saying that sleeping is your main "activity" at night:

La nuit, je dors.