You and your son were lucky; my son, also born in TPMH but some 20 years earlier was also refused permission to enter the US when travelling on his British passport as he had put RAF Akrotiri, Cyprus as his place of birth in the ESTA and was therefore also ineligible for the Visa waiver programme. After a fairly intense interrogation he was imprisoned for 24 hours then deported - handcuffs and manacles to the airport, and 'restrained' on the aircraft until clear of US territory
That's curious, because I have been entering the US several times per year for the last 16 years, with entry points East, central and Western, and never had a query or problem. Looking at my passport right now, I see I have the place of birth as simply "RAF Akrotiri" so perhaps that's part of the answer.