1. A King is born, He comes to save the Lord
Why December 25?
A King is born, He comes to save the Lord

2. This is a complex Question
The Source from the online Catholic Encyclopedia gives a thorough analysis:
Between 337 AD and 352 Pope Julius set the celebration of Christmas on December 25.
In the East the feast of the Epiphany, January 6, is often celebrated as the birth of Christ.

3. The Counter Claims
Mithraism and Sol Invictus (the unconquerable sun) really took of about 100 years after the Death of Jesus. It seems that the pagan elites may have even used this myth to counter the growing influence of Christianity
The Winter Solstice, clearly this is when the night is the longest and the “day” starts to make gains over the night. The symbolism of Jesus (the Son) conquering the darkness is clearly evident, but it is impossible to determine if this was the predominant factor.
Saturnalia was a pagan festival that was celebrated in early December to the 23rd. Most scholars seem to dismiss any serious connection.

4. Claims from Census Records
No records currently are known, but several of the Church Fathers make mention of them. Their claims seem to be plausible in light if the organization of the Roman Empire.
Justin Martyr ( AD) and Tertullian ( AD) mention the census records in Rome, along with St. John Chrysostom ( AD).
St. Cyril is to have written a letter to Pope Julius to confirm the date by referencing the census documents.

5. Looking at Biblical Clues
From the Bible we know that Zechariah was from the temple priests for the clan of Abiha, and some claim to make the reckoning of the date by the time of his temple service.
Elizabeth is assumed to have conceived some short time after his temple service, and we know Mary conceived 6 months after Elizabeth’s conception, which by some reckoning places the birth of Jesus near December.

6. Astronomical Phenomenon
There is a compelling website: this site recounts some signs of the times that would have been apparent in the sky over ancient Jerusalem that would have been visible to the kings.
There is a conjunction of the planets Jupiter, Venus, and the star Regulus on December 25, 2 BC.

7. The Real Point
With our current records, there is no way to be certain regarding the date of the birth of Jesus, but it is important to note that what truly matters is that Jesus was born, the Word became flesh!
Unlike Mithras or the myths of various pagan traditions, Jesus is not a myth, HE IS!