One of the arguments in favor of the traditional date of December 25 being the actual date of Jesus birth is that of church tradition.
Of course, it's possible that the early church just picked a date connected with the winter solstice to celebrate the birthday of Jesus. The scholar Ray Butterworth points out how the Queen's Birthday, is celebrated on the second Saturday in June in the UK, and the Monday before May 25 in Canada, regardless of her actual birthday.
However, it is plausible (perhaps even probable) that Mary, the mother of Jesus, might have spoken to the disciples in the early faith communities about the specific symmetrical period and days of Jesus' conception and birth.
While one might argue that was improbable that there was a perfect symmetrical time of 9 months between the conception of Jesus and his birth, C.S. Lewis made a compelling observation:
You must develop a nose like a bloodhound for those steps in the argument which depend not on historical and linguistic knowledge but on the concealed assumption that miracles are impossible, improbable, or improper. And this means that you must really re-educate yourself: must work hard and consistently to eradicate from your mind the whole type of thought in which we have all been brought up. (Miracles)
A good discussion on stackexchange on the matter of whether it was appropriate for Jews to celebrate birthdays can be found here. The quote from Rabbi Mendy Kaminker, in support of Jews celebrating birthdays, is particularly relevant.
The first century philosopher Plutarch writes:
...they are those that tell us that, as the Greeks are used to allegorize Kronos (or Saturn) into chronos (time), …
Lucian of Samosata (AD 120-180) describes the festivities of Saturnalia/Chronos, as seven day event. Marcus Valerius Martialis in his also speaks of the Saturnalia being a seven day event (Epigrams, 72, Book XIV). These authors wrote prior to when the Sol Invictus was highlighted and celebrated as a Roman holiday in the fourth century.
Changes to the Roman calendar moved the climax of Saturnalia/Chronos to December 25th, around the time of the date of the winter solstice.
So, if December 25th was the birthday of Jesus that Mary shared with the disciples, it may very likely be a type of double entendre when the Apostle Paul writes:
...when the fullness of time (chronos) had come, God sent forth his Son, born of woman, born under the law, to redeem those who were under the law,... (Galatians 4)
In other words, the primal origins of Christmas can be found in a rebranding of Saturnalia/Chronos in line with longings of a mythical golden age coming in Jesus in terms of both time (chronos) and spatial (literal) history.
In scanning the internet I have come across multiple references to Theophilus (A.D. 115-181), bishop of Caesarea in Palestine:
We ought to celebrate the birthday of Our Lord on what day soever the 25th of December shall happen. - Magdeburgenses, Cent. 2. c. 6. Hospinian, De origine Festorum Chirstianorum.
The problem is that I can't find the original source for this quote. Has it been lost in history? If so, what are some plausible reasons on how this might have happened? In researching the background to the quote from Theophilus, the evidence so far indicates that this quote seems to be based on a fictional account created in Ireland around 600 AD. For example, see this recent research by Roger Pearse.
In another line of tradition, McClellan notes that Chrysostom around 407 A.D. (Hom. in diem, natal, ii Col. 351) writes:
It is not yet ten years since this day (December 25) was made known. Even so, it is now just as seriously observed as if it has come to us form the the beginning...It is very plain, according to the evangelist, that Christ was born during the first census, and in Rome it is possible for any one to deduce, with the aid of the public archives, when this came about. From persons who have an intimate knowledge of the records and who still live there and who have kept the day in accordance with an age long tradition have recently given us this information. (McClellan, p. 407)
Could a large section of the church in those days have forgotten the date of Jesus' birth so easily? If so what are some plausible explanations? Was Chrysostom somewhat limited in his information from other areas? For example, he writes in another place that the charismatic gifts ceased. Yet, other church fathers attest to their continuance.
An argument for Clement (150–215 AD) of Alexandria (Stromateis 1.21.145) saying that some hold to December 25th as Jesus' birthday can be found here.
The author argues that Clement's 25th day of Pachon should be interpreted along the lines how, "When viewed through the prism of the Athenian calendar, we find that the traditional date of December 25 comes forth quite naturally."
Another site, here, argues that Clement held to January 6th as the birthdate of Jesus:
Can we pinpoint the date of Jesus' birth? Various ancient sources lead us to the answer. Clement of Alexandria (again, about AD 200) says, "From the birth of Christ, therefore, to the death of Commodus ...are, in all, a hundred and ninety-four years, one month, thirteen days." If we suppose that he is using the Roman calendar, we deduce that Clement set Christ's birth on 18 November 3 BC. But it is highly doubtful that this date, affirmed by no other ancient source, is the one he so confidently espouses. We arrive at a different date if we suppose that Clement, a resident of Egypt, is using the Egyptian calendar without intercalation. Measuring backward from Commodus' death an interval of 194 years (each exactly 365 days), one month (thirty days), and thirteen days brings us to 6 January 2 BC.
There is also a mentioning of the December 25th by Hippolytus in his commentary on Daniel (dated 202 A.D.). The quote itself, along with a textual critical approach on the reliability of this reference, can be found here.
The total references, included above, push the earliest testimony of the church fathers to the date of Jesus birth on December 25th to the late 2nd century and early 3rd century.
A related question can be found here: What is the earliest historical testimony of the celebration of the nativity of the Lord Jesus Christ?
I follow what C.S. Lewis once said, "You must develop a nose like a bloodhound for those steps in the argument which depend not on historical and linguistic knowledge but on the concealed assumption that miracles are impossible, improbable, or improper."
– Jess Dec 23 '21 at 20:26