The baby's name was Cooper. He wasn't credited in the cast list because it's not a speaking role. We hear his name only twice that I recall - when Mason Sr. mentions to the kids that it means a lot to Annie for them to come to Cooper's baptism. Then another time when Mason is on the phone to Mason Sr., who is with Annie and the baby, and asks Mason Jr. to "say Hi to Annie and Cooper," or something like that.
We don't see Cooper grow up because the film, originally titled "Twelve Years," was to cover the time span between Mason Jr.'s first grade year and his entry to college. At that point Cooper was still a baby.