I'm looking into the tradition of christmas and was wondering if we actually know the date of his birth and have evidence for it.
The short answer is we don't have a clue.
The reason why the Church celebrates His birth on December 25th goes back to the early centuries of Christianity. Early Christians were interested in trying to figure out when Christ was born, and there were several different thoughts and opinions at the time:
1. Some Christians, such as Origen of Alexandria, considered it inappropriate to celebrate Christ's birth at all, with the opinion that "only pagans celebrate birthdays".
2. Most Christians at the time were convinced that Christ had been crucified on a March 25th, and so they reasoned that a perfect person had a perfect life, and thus if He had been crucified on March 25th then He was either born or conceived on March 25th. And, so yes some Christians did celebrate His birth on March 25th, others however believed He was conceived on that date, and thus added nine months, and thus celebrated His birth on December 25th.
3. Many other Christians believed that the celebration of His birth should be part of the already existing Feast of Epiphany (also called Theophany), which is on January 6th. Even today the ancient Armenian Church continues to celebrate Christ's birth on January 6th, not December 25th.
Sometime in the 5th century the timing of Christmas was standardized, and December 25th was chosen as the official date to celebrate Christ's Nativity, it was at that time the most popular date anyway.
There is some debate as to how influential the Pagan Dies Natalis was on choosing December 25th for Christmas, because in the 3rd century, the same time as Christians were trying to figure out when Jesus was born and when to celebrate it, the Romans created a new religion dedicated to the worship of Sol Invictus (the Unconquered Sun) whose birth feast (Dies Natalis) was celebrated on December 25th, coinciding with the old winter solstice. Some have suggested Christmas is merely a rip off of the Dies Natalis, but I haven't found any reason to believe this especially since Christians were still working out when to celebrate Christmas even into the 5th century. But it is possible, that as Christianity and Paganism continued to butt heads, and as the 4th century saw the rise of Christianity as the favorable religion in the Roman Empire and Paganism lost its favor, that having the celebration of Christ's birth on December 25th made it less difficult for pagans who converted to Christianity who no longer worshiping the sun now began worshiping the Son.
The reality is that the historical record doesn't seem to be particularly clear, and there tends to be a lot of guesswork; but it is safe to say that Christmas exists as an independent Christian religious observance apart from Pagan observances, but that it's possible that the popularity of the Pagan observance helped push the Church toward a more universal acceptance of December 25th as a way to supplant and replace Pagan observances with Christian ones--but that's merely conjecture. Christmas, however, remains a thoroughly and distinctively Christian holiday and season.
And it's worth emphasizing that even if Jesus was born in the summer, or in the spring, Christmas isn't really about celebrating Jesus' "birth day", but celebrating
that He was born, period.
-CryptoLutheran