I looked up the thing about Lydia being a daughter of St. Joseph — indeed a passage called “The History of Joseph the Carpenter” in one of apocryphal books known as the Arabic Gospel of the Infancy she is listed, along with sister Assia and brothers Judas, Justus, James, and Simon, as the child of St. Joseph from his first marriage (second paragraph).

Regarding this particular book (from New Advent):

“Arabic Gospel of the Infancy

The Arabic is a translation of a lost Syriac original. The work is a compilation and refers expressly to the “Book of Joseph Caiphas, the High-Priest”, the “Gospel of the Infancy”, and the “Perfect Gospel”. Some of its stories are derived from the Thomas Gospel, and others from a recension of the apocryphal Matthew. However there are miracles, said to have occurred in Egypt, not found related in any other Gospel, spurious or genuine, among them the healings of leprosy through the water in which Jesus had been washed, and the cures effected through the garments He had worn. These have become familiar in pious legend. So also has the episode of the robbers Titus and Dumachus, into whose hands the Holy Family fell. Titus bribes Dumachus not to molest them; the Infant foretells that thirty years thence the thieves will be crucified with Him, Titus on His right and Dumachus on His left and that the former will accompany Him into paradise. The apocryphon abounds in allusions to characters in the real Gospels. Lipsius opines that the work as we have it is a Catholic retouching of a Gnostic compilation. It is impossible to ascertain its date, but it was probably composed before the Mohammedan era. It is very popular with the Syrian Nestorians. An originally Arabic “History of Joseph the Carpenter” is published in Tischendorf’s collection of apocrypha. It describes St. Joseph’s death, related by Our Lordto His disciples. It is a tasteless and bombastic effort, and seems to date from about the fourth century.”

Also this (from here on New Advent):

“It will not be without interest to recall here, unreliable though they are, the lengthy stories concerning St. Joseph’s marriage contained in the apocryphal writings. When forty years of age, Joseph married a woman called Melcha or Escha by some, Salome by others; they lived forty-nine years together and had six children, two daughters and four sons, the youngest of whom was James (the Less, “the Lord’s brother”). A year after his wife’s death, as the priests announced through Judea that they wished to find in the tribe of Juda a respectable man to espouse Mary, then twelve to fourteen years of age. Joseph, who was at the time ninety years old, went up to Jerusalem among the candidates; a miracle manifested the choice God had made of Joseph, and two years later the Annunciation took place. These dreams, as St. Jerome styles them, from which many a Christian artist has drawn his inspiration (see, for instance, Raphael’s “Espousals of the Virgin”), are void of authority; they nevertheless acquired in the course of ages some popularity; in them some ecclesiastical writers sought the answer to the well-known difficulty arising from the mention in the Gospel of “the Lord’s brothers”; from them also popular credulity has, contrary to all probability, as well as to the tradition witnessed by old works of art, retained the belief that St. Joseph was an old man at the time of marriage with the Mother of God.”

So they’re “void of authority,” but definitely interesting!