It seems that the Roman emperor Constantine is credited with picking 25 December after he converted in 313AD following a military victory after being persuaded to paint crosses on his soldiers' shields. He might have opted for 25 December because it was the date when the birth of Mithras was celebrated and not long after celebrations for Saturn (17 December). Giving presents on these days was familiar in Persian and Roman culture.
In 376AD, Pope Leo destroyed the temple of Mithra, but the Mithraic festivity of the birth of the Sun continued, largely because it was a convenient time to be merry in the middle of the winter. It was not till the year 530AD that the church commissioned the monk Dionysius Exiguus to proclaim this popular festivity as the birth of Christ.
The New Schaff-Herzog Encyclopedia of Religious Knowledge explains:
"How much the date of the festival depended upon the pagan Brumalia (December 25th) following the Saturnalia (December 17th-24th), and celebrating the shortest day of the year and the 'new sun' ... cannot be accurately determined. The pagan Saturnalia and Brumalia were too deeply entrenched in popular custom to be set aside by Christian influence ... The pagan festival with it's riot and merrymaking was so popular that Christians were glad of an excuse to continue its celebration with little change in spirit and in manner. Christian preachers of the West and the Near East protested against the unseemly frivolity with which Christ's birthday was celebrated, while Christians of Mesopotamia accused their western brethren of idolatry and sun worship for adopting as Christian this pagan festival."
Midwinter was celebrated by many cultures. In Babylon it was dedicated to the Queen of Heaven, in Greece to Dionysos and Adonis, in Rome it was to Saturn (as mentioned above), in England to the Nordic god Balder.
St Augustine apparently said: "we hold this (Christmas) day Holy, not like the pagans because of the Birth of the Sun, but because of the birth of him who made it (Christmas) day Holy, not like the pagans because of the Birth of the Sun, but because of the birth of him who made it", which shows that it was a festival that was dedicated to the birth of the sun that was adopted by the Christians.
Apart from Christmas, Christianity also absorbed many Mithraic myths and legends like immaculate birth, baptism, Eucharist, 12 disciples, birth in a cave/stable, the presence of shepherds at the time of the birth and the resurrection. Like the creation myth, these are not unique to Christianity.