In the wake of The da Vinci code and claims about a Jesus family tomb including a reference to the son of Jesus, some people actually think there is some merit to the idea that Jesus was married and had a child. That there is not a shred of tradition or evidence in any ancient Christian writings for the existence of a wife and child seems to be beside the point.
Consider what would be the case if Jesus had a son. Most of his followers considered Jesus the Son of God and a deity himself. Any child would then also have been thought of as the Son of God and a deity and would have been a chief figure and icon in Christian circles. He would have been given a central role in the Church movement and been widely revered as his father's successor. Yet, despite the efforts of early Christians to know everything they could about Jesus, with many mythological fabrications and other writings circulating alongside accepted Church writings, none reference a son of Jesus.
One person who would most likely have known about any alleged son of Jesus would have been the Church historian Eusebius who attempted to write as thorough a history as he could of the Christian movement down to this time in the fourth century. He cited numerous traditions that had been handed down and one topic of concern for him was the family of Jesus.
So, what did Eusebius have to say after investigating all the written sources and traditions he knew about?
When James, the brother of Jesus and first head of the Church (over Peter and Paul), was martyred (in Year 62, per Josephus) and after the fall of Jerusalem (Year 70), there was a conclave of all the living Apostles and disciples "together with those who, humanly speaking, were kinsman of the Lord-for most of them were still living." They selected "Symeon, son of the Clopas mentioned in the Gospels" to serve on the throne of Jerusalem, and become successor of James. (EH 3.11.1.)
As Clopas was apparently a brother of Jesus' father, Symeon was Jesus' cousin. If ever there was a time when a tradition about a son and a wife of Jesus would arise, if either existed, surely this convocation of living family members and apostles would have been such an occasion. But apparently, a cousin was the closest they could come up with to serve as leader of the Church.
Now, whether this story is true or not is not the issue. What mattersis that there is not the slightest hint here of any tradition concerning a son or wife of Jesus.
In a further investigation of Jesus' surviving family members, apparently the most Eusebius could come up with, for all practical purposes, was that Jesus had a brother named Jude (attested in the Gospels) and this Jude had some grandchildren, and towards the end of the first century, under the Emperor Domitian, they were questioned as possible descendants of David and, therefore, potential enemies of Rome. Domitian decided they represented no threat and let them go. On their release, they became leaders of the Church. (EH 3.20.)
So, all our earliest traditions can come up with are a couple of nephews and a cousin, with what strikes me as an overwhelming circumstantial case that Jesus had no son.