It's often suggested that the authors of the infancy narratives drew some of their material from earlier sources. As far as earlier sources affirmed something like the virginal conception, the Bethlehem birthplace, or the Slaughter of the Innocents, Matthew and Luke aren't alone in affirming such things. Critics can't propose multiple layers of development behind the infancy narratives, with earlier sources making claims that Matthew and Luke repeated, yet turn around and object in other contexts that Matthew and Luke are the only early sources who made the claims in question. As Charles Quarles notes regarding the notion that multiple sources behind Matthew's gospel affirmed the virginal conception:
That allusion or affirmation of the virginal conception appears in multiple pre-Matthew sources should make one pause before dismissing it too lightly. (in Robert Stewart and Gary Habermas, edd., Memories Of Jesus [Nashville, Tennessee: B&H Publishing Group, 2010], approximate Kindle location 4168)
Critics can't have it both ways. They can't argue, on the one hand, that Matthew and Luke were alone in making particular claims, yet argue, on the other hand, that they (Matthew and Luke) received those claims from earlier sources.
It's often suggested that extrabiblical Christian sources who repeat what's found in the Biblical documents must merely be repeating what they derived from those documents. But we need to be critical of that suggestion. Ancient Christians lived in an environment much closer to the Biblical context than we are. They had access to many documents that are no longer extant, for example. Men like Clement of Rome and Ignatius were alive as adults in the first century and were contemporaries of the apostles. Even in later generations, think of what a source like Justin Martyr tells us about the information he has. He refers, for example, to a government record of the census of Luke 2 (First Apology, 34), a Bethlehem cave in which Jesus was born (Dialogue With Trypho, 78), the location the magi came from (Dialogue With Trypho, 78), and the type of woodworking Jesus did as a carpenter (Dialogue With Trypho, 88). He didn't get any of that information from the New Testament. Justin was aware of many Jewish sources and traditions outside the Bible, as I've discussed elsewhere. Why should we think Justin was just uncritically repeating what he read in the Biblical documents when he addressed Jesus' childhood? Much the same can be said of other Christians writing outside the New Testament. Though dependence on the Biblical documents does diminish the evidential value of the early extrabiblical Christian sources to some extent, it doesn't eliminate their value. When so many Christians of the patristic era affirm Jesus' Davidic ancestry, his Bethlehem birthplace, Luke's census, and other such things, those affirmations carry some evidential weight.
In addition to not being too dismissive of the testimony of the early Christians, we need to be careful not to neglect heretical sources. Just as ancient Jews and pagans had reason to be critical of the infancy narratives and other early sources making claims about Jesus' childhood, so did many ancient heretics. Some heretics rejected Matthew and Luke or one of the two. Or they accepted much of Luke, but not his infancy account. When such sources corroborate early Christian claims about Jesus' childhood, and do so against their own interests, that's significant. And the testimony of heretics in general, even when not acting against their interests, has the same sort of value as the testimony of other historical witnesses.
An important issue in evaluating the early claims made about Jesus' childhood is the authorship of the documents involved. What did the early Christian, heretical, Jewish, and pagan sources claim about the authorship of the New Testament, for example? See here, here, here, and here.
Early in the second century, Ignatius refers to how Jesus was "truly" a descendant of David and born of a virgin, contrasting such a view of Jesus with what some heretics believed (Letter To The Smyrnaeans, 1-2). Apparently, the Docetists Ignatius is referring to claimed that Jesus only appeared to have the physical attributes referred to in the New Testament (his birth, death, resurrection, etc.). These heretics had an aversion to the material world. If the Docetists thought they had to acknowledge the existence of an appearance that Jesus was involved in such things, that concession has significant evidential value. Instead of denying that the events even appeared to occur, they thought they had to admit such an appearance.
As I noted in some earlier posts, we have evidence from the New Testament for an early Jewish claim that Jesus was born illegitimately. There are traces of that claim in early extrabiblical literature as well. The dispute over how Jesus was conceived and other arguments about Jesus' childhood are prominent in the early interactions between Christians and Jews. I'll give some examples.
In early Jewish tradition, sometimes Mary is named and her ancestry is alluded to. W.D. Davies and Dale Allison mention some of the sources (Matthew 1-7 [New York, New York: T&T Clark, 2010], n. 73 on 184). Craig Keener writes, "B. Sanh. 43a, bar., may preserve a [non-Christian Jewish] tradition that Jesus was of royal lineage (unless it suggests connections with the Herodian or Roman rulers, or that he was about to take control of the people; both views are unlikely)." (A Commentary On The Gospel Of Matthew [Grand Rapids, Michigan: Eerdmans, 1999], n. 9 on 75) The Jewish charge of sexual immorality against Mary is repeated by Celsus in his work against Christianity in the late second century (Origen, Against Celsus, 1:28, 1:32). For further discussion, see Henry Chadwick, ed., Origen: Contra Celsum (New York, New York: Cambridge University Press, 2003), n. 3 on 31.
Jesus' presence in Egypt for part of his life is widely corroborated by Jewish and pagan sources. See the following for a discussion of those sources: John Cook, The Interpretation Of The New Testament In Greco-Roman Paganism (Peabody, Massachusetts: Hendrickson Publishers, 2002), 32-33; W.D. Davies and Dale Allison, Matthew 1-7 (New York, New York: T&T Clark, 2010), 259-260. Since both pagan and Jewish opponents of Christianity had an interest in dismissing Jesus' miracles as magic he learned in Egypt, the significance of pagan and Jewish corroboration of a sojourn in Egypt is diminished. But it does have some significance. Jesus could have been accused of magic without a sojourn in Egypt. And just as accusing Jesus of learning magic in Egypt would have had some appeal to early enemies of Christianity, so would contradicting Matthew 2 by denying that Jesus was in Egypt at all.
Though the New Testament and other sources suggest that some of the Jewish arguments against Christianity were formed early, much of what the Jewish sources say is late, garbled, or problematic in some other way. As Keener notes, "later rabbinic tradition about Mary (properly 'Miriam'; see Dalman 1973: 31) probably depends on and garbles early Christian sources." (A Commentary On The Gospel Of Matthew [Grand Rapids, Michigan: Eerdmans, 1999], 85) Still, it would be unreasonable to argue that the earliest Jewish opponents of Christianity were highly apathetic and/or ignorant about Christianity and didn't independently preserve any information about the events surrounding Jesus' birth. It's significant that there's agreement between early Jewish and Christian sources on matters like Jesus' ancestry and the premarital timing of his conception.
For a discussion of patristic Christian sources on Jesus' Bethlehem birthplace, see here. Regarding early non-Christian sources, see here.
I wrote an article a few years ago that discusses what early non-Christian sources say about the Slaughter of the Innocents.
A series of posts I wrote about the census of Luke 2 addresses how early Christian and non-Christian sources viewed the historicity of the passage.
Celsus and his Jewish source(s) say that Jesus claimed he was born of a virgin (in Origen, Against Celsus, 1:28). Notice the pagan and Jewish acceptance of the earliness of the virgin birth claim. It's attributed to Jesus himself. Similarly, Celsus attributes other infancy narrative claims to Jesus (ibid., 1:58). Contrast that to the tendency of modern critics to speculate that these claims about Jesus' childhood didn't arise until later.
Notice how widespread these Christian beliefs about Jesus' childhood were. Note how widely the early opponents of Christianity attributed such beliefs to Christians and sometimes even accepted the beliefs themselves. I recently discussed some of the evidence to that effect with regard to the virgin birth, for example: see here and here. If concepts like the virginal conception and Jesus' birth in Bethlehem were unpopular early on, and men like Paul, Mark, and John had spent decades teaching against such beliefs, then the widespread nature of those beliefs in the earliest centuries is difficult to explain.
To appreciate the significance of what I've outlined in this post, think of what early Christian history might have looked like instead. Why are so many of the theories of modern critics of the infancy narratives either absent or highly unpopular, including in non-Christian circles, in the earliest centuries? If Jesus was born in Nazareth or Luke's census account is as wrong as modern critics often suggest, for example, why do those modern critical theories have such a lack of corroboration not only in ancient Christian sources, but in ancient non-Christian sources as well? Critics of the infancy narratives often object to a supposed lack of corroboration of the infancy accounts in other sources. Not only is that objection inaccurate, as I've argued in this series, but it's also more applicable to the critic's position than the Christian's.