Dies Natalis Solis Invicti means “the birthday of the unconquered sun”. Some early Christian writers connected the sun to Jesus, whom they believed was prophesied in Malachi 4:2 as the “Sun of Righteousness”: “O, how wonderfully acted Providence that on that day on which that Sun was born…Christ should be born”, a work attributed to Cyprian said.   
However, this is unrelated to the winter solstice and is instead based on the idea that creation began on 25 March (the first day of the week, Sunday) and that Jesus was born on 28 March, the day of the creation of the sun (the fourth day of the week, Wednesday): “O the splendid and divine providence of the Lord, that on that day, the very day, on which the sun was made, the 28 March, a Wednesday, Christ should be born. For this reason Malachi the prophet, speaking about him to the people, fittingly said, ‘Unto you shall the sun of righteousness arise, and healing is in his wings.'”
In the fourth century, John Chrysostom, promoting celebration on 25 December, commented on the connection: “But Our Lord, too, is born in the month of December … the eighth before the calends of January [25 December] …, But they call it the ‘Birthday of the Unconquered’. Who indeed is so unconquered as Our Lord …? Or, if they say that it is the birthday of the Sun, He is the Sun of Justice.” With regard to a supposed December religious feast of the sun as a god (Sol), rather than a solstice feast of the physical sun, however, “while the winter solstice on or around December 25 was well established in the Roman imperial calendar, there is no evidence that a religious celebration of Sol on that day antedated the celebration of Christmas”.