Is there a synonym for a car's interior? E.g. 'cockpit', 'cabin', etc. "He sat quietly in the _____ and waited."
The word that you are looking for is called passenger seat.
Merriam Webster defines it as
the front seat of a vehicle (such as a car) where a passenger sits
Also, we can cross verify it with Oxford Dictionary.
A seat provided in a vehicle for a passenger; specifically (in a motor vehicle) the seat next to the driver
"His car was released to his fiancée, who was riding in the passenger seat and was sober, Paul said."
"He slouched into the passenger seat of the ivory Ford sedan and shut the door hard"
In your example sentence, just 'car' works fine. The preposition 'in' already makes clear that the subject is not on the roof or the bonnet.
That said, according to Merriam Webster, a cockpit is
4 : a space or compartment in a usually small vehicle (such as a boat, airplane, or automobile) from which it is steered, piloted, or driven
(emphasis mine) so it's applicable here as well.
If you need to specify the position, consider
He sat quietly on the driver's/passenger's/rear seat and waited.