The Church has celebrated Mary's birth since at least the sixth century. A September birth was chosen because the Eastern Church begins its Church year with September. The September 8 date helped determine the date for the feast of the Immaculate Conception on December 8 (nine months earlier).
Scripture does not give an account of Mary's birth. However, the apocryphal Protoevangelium of James fills in the gap. This work has no historical value, but it does reflect the development of Christian piety. According to this account, Anna and Joachim are infertile but pray for a child. They receive the promise of a child that will advance God's plan of salvation for the world. Such a story (like many biblical counterparts) stresses the special presence of God in Mary's life from the beginning.
St. Augustine connects Mary's birth with Jesus' saving work. He tells the earth to rejoice and shine forth in the light of her birth. "She is the flower of the field from whom bloomed the precious lily of the valley. Through her birth the nature inherited from our first parents is changed." The opening prayer at Mass speaks of the birth of Mary's Son as the dawn of our salvation and asks for an increase of peace.
We can see every human birth as a call for new hope in the world. The love of two human beings has joined with God in his creative work. The loving parents have shown hope in a world filled with travail. The new child has the potential to be a channel of God's love and peace to the world.
This is all true in a magnificent way in Mary. If Jesus is the perfect expression of God's love, Mary is the foreshadowing of that love. If Jesus has brought the fullness of salvation, Mary is its dawning.
Birthday celebrations bring happiness to the celebrant as well as to family and friends. Next to the birth of Jesus, Mary's birth offers the greatest possible happiness to the world. Each time we celebrate her birth we can confidently hope for an increase of peace in our hearts and in the world at large.