I would say "the couple has a son" because you are referring to the couple as a unit, not the individual members. (Think "The crowd chants", not "The crowd chant".) "The couple have sons" would make sense (you can think of each member of the couple having a son), but even that is a stretch.
Guess it depends on context! "The combo of female, deadpan drummer and drawling guitar-singer has been heavily imitated since, but certainly stood out when their album launched. The pair have a son together, but their relationship is now firmly artistic."
Generally agree though.
It's a difference I've noticed between American English and British English. In Am-Eng a noun that denotes a group is often treated as a singular entity, so the verb is conjugated accordingly. But Br-Eng will conjugate the verb as a plural, reflecting the multiple "parts" that make up the noun.
Ex, American English: -The council has announced... -The family has made...
Ex, British English: -The council have announced... -The family have made...