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While there is no real scientific way to determine the date of Jesus's birth, there is ample reason to believe that it could indeed have occurred on Dec. 25th...
Firstly, Luke mentions (1:26) that "in the sixth month" the angel Gabriel visited Mary to tell her she was going to bear a son. The sixth month of the old Hebrew calendar could have been right around April 1st (this is bolstered by the fact that Elizabeth, her cousin, was about 6 months pregnant with John who was conceived, according the Scripture, sometime in the autumn of the previous year.
Secondly, Joseph and Mary were traveling to Bethlehem to be counted in the census for tax purposes...and, since they were under Roman rule, it would make logistical sense that the Roman government would require the census to begin the first month of the Roman calendar.
Thirdly, there is the astronomical data that puts the star of Bethlehem (in the constellation of Aries) at its peak in the winter.

I have heard various accounts of evidence that Christ had to be born in the spring as that was when shepherds would be watching their flocks at night as lambs would be born in the early spring...however, the Bible does not specify exactly when they came to the manger. I reason it could have been late February or early March--when the lambs are usually born.

Gospel Records of Jesus' Birth -- Bible Discussion

The question is whether you believe and trust
the Holy Word of Elohim in Luke 1
or you trust the traditions of man

Yah'shua's birth on Sukkot (September 26 at sundown to October 3 2007)
(Sukkot is the Feast of Tabernacles or booths, where we live in temporary shelters.
Sukkot is when YHvH took on a temporary garment to be with His People
and to die as the Lamb of G-d on Pesach in order to bring salvation to all
who would call on His Name : Yah'shua ( YHvH is become my salvation)).
Sukkot as the date is supported by Elizabeth's pregnancy of John the Immerser.
The time sequence is outlined by the Holy Word of Elohim in Luke 1 with Zacharias.
Zacharias served as a high priest and based on his tribe, we know when he served
and when he was struck dumb and when John was conceived.
John would have been born on Pesach. Most Jews believed that
Elijah would come at Pesach to announce the coming of the Messiah
Factor in when Miriam visited her cousin Elizabeth,
Elizabeth was six months pregnant (Luke 1:26)
Thus the timing of Yah'shua's birth can be ascertained.
John (1:14) tells us that Yah'shua was made flesh and tabernacled among us.
Eight days after the beginning of Sukkot is another Holy Feast Day called Shemini Atzeret.
Eight days after a Jewish male is born he is circumcised.
Nine months back from Sukkot is Chanukah where the light entered the temple.

b'SHEM Yah'shua

Interesting facts, especially about the Star of Bethlehem and annunciation date.

Then we have the quote from John the Baptist that has been aligned allegorically with the calendar — “He must increase; I must decrease.”

Christ was born in the winter when the days start to get longer — increase. John the Baptist was born in the summer when the days start getting shorter — decrease.