Does my unvaccinated 16-year-old son need a COVID test to fly with his vaccinated Thai mother to Bangkok?

This is from the tatnews website, that always has the most recent update to rules.

/thailands-entry-requirements/

Under the vaccinated traveler section:

Travellers 5-17 years of age travelling to Thailand unaccompanied must get vaccinated with at least 1 dose of an approved vaccine at least 14 days before travelling to Thailand. Those travelling with parents are exempt from this requirement.

Less than 5 years old, no vaccine or test required at all.

5 to 17 years old, no vaccine or test required if accompanied by a fully vaccinated adult.

I believe he will need a covid test done by a lab, or clinic in order to enter as he is unvaccinated.

That's what the current rules say. Under 18 and flying with a vaccinated parent, the child does not require vaccination.