If you are looking for an answer that says "We know that Jesus was really born on December 25th and here is the evidence", you are going to be disappointed. There is no such evidence, and we don't know his birthday. (There is some circumstantial evidence that it was in Spring, but it's far from conclusive). December 25th was indeed arbitrarily chosen, and probably deliberately to coincide with popular midwinter festivals.
There is however no reason why this should be taken to invalidate any part of the Christian faith. Nothing in Christian belief requires us to know Jesus' birth date, and there is nothing to say that if we choose to celebrate his birth then it has to be on a day when nobody else is celebrating anything. If you are going to celebrate his birth (and it's surely worth celebrating) then you have to pick one day or another.
You might like to compare this with people who don't know their own birthday. That's certainly no reason to disbelieve their existence, and quite frequently people in that position choose an arbitrary date to celebrate their birth. They might well choose a convenient date, possibly one in which another celebration is already going on. That would not make the celebration, or the birth, less valid.
Alternatively, you might think of it as an "official birthday", like those celebrated by Queen Elizabeth of the United Kingdom - it's isn't actually her birthday, it's just a special time set aside to celebrate.