In Scandinavia (as late as until the mid 18th century) this date was the longest night of the year, coinciding with Winter Solstice, due to the Julian Calendar being employed at that time.
The same can be seen in the poem "A Nocturnal upon S. Lucy's Day, Being the Shortest Day" by the English poet John Donne.

While this does not hold for our current Gregorian calendar, a discrepancy of 8 days would have been the case in the Julian calendar during the 14th century, resulting in Winter solstice falling on 13 December.
With the original adoption of the Gregorian calendar in the 16th century the discrepancy was 10 days and had increased to 11 days in the 18th century when Scandinavia adopted the new calendar, with Winter solstice falling on 9 December.

The Winter solstice is not visibly shorter than the several days leading up to and following it and although the actual Julian date of Winter solstice would have been on 15 or 14 December at the time when Christianity was introduced to Scandinavia, 13 December could well have lodged in peoples mind as being the shortest day.

The choice of 13 December as Saint Lucy's day, however, predates the eight-day error of the 14th century Julian calendar.
This date is attested in the pre-Tridentic Monastic calendar, probably going back to the earliest attestations of her life in the 6th and 7th centuries, and it is the date used throughout Europe.
So, while the world changed from a Julian to a Gregorian calendar system—and hence acquired a new date for the Winter Solstice—St Lucy's Day was kept at 13 December, and not moved to the 21.

In the Roman Empire, 25 December (in the Julian Calendar) date was celebrated as being the day when the Sun was born, the birthday of Sol Invictus, as can be seen in the Chronography of 354.
This date corresponding to the date of the Winter solstice.
Early Christians considered this a likely date for their saviour's nativity, as it was commonly held that the world was created on Spring equinox (thought to fall on 25 March at the time), and that Christ had been conceived on that date, being born 9 months later on Winter solstice.

A Swedish source states that the date of (Winter Solstice, St. Lucia, Lucinatta, Lucia-day, Lussi-mass ...) i.e. 13 December, predates the Gregorian which implies that "Lucia's Day" was 13 Dec in the Julian Calendar, which is equal to 21 December in the Gregorian, i.e. now.
Same source states use of the name "Little Yule" for the day, that it was among the most important days of the year, that it marked the start of Christmas month, and that with the move to the Gregorian calendar (in Sweden 1753) the date (not the celebration) "completely lost its appropriateness/significance".
In Scandinavia (as late as until the mid 18th century) this date was the longest night of the year, coinciding with Winter Solstice, due to the Julian Calendar being employed at that time. The same can be seen in the poem "A Nocturnal upon S. Lucy's Day, Being the Shortest Day" by the English poet John Donne.
While this does not hold for our current Gregorian calendar, a discrepancy of 8 days would have been the case in the Julian calendar during the 14th century, resulting in Winter solstice falling on 13 December. With the original adoption of the Gregorian calendar in the 16th century the discrepancy was 10 days and had increased to 11 days in the 18th century when Scandinavia adopted the new calendar, with Winter solstice falling on 9 December.
The Winter solstice is not visibly shorter than the several days leading up to and following it and although the actual Julian date of Winter solstice would have been on 15 or 14 December at the time when Christianity was introduced to Scandinavia, 13 December could well have lodged in peoples mind as being the shortest day.
The choice of 13 December as Saint Lucy's day, however, predates the eight-day error of the 14th century Julian calendar. This date is attested in the pre-Tridentic Monastic calendar, probably going back to the earliest attestations of her life in the 6th and 7th centuries, and it is the date used throughout Europe. So, while the world changed from a Julian to a Gregorian calendar system—and hence acquired a new date for the Winter Solstice—St Lucy's Day was kept at 13 December, and not moved to the 21.
In the Roman Empire, 25 December (in the Julian Calendar) date was celebrated as being the day when the Sun was born, the birthday of Sol Invictus, as can be seen in the Chronography of 354. This date corresponding to the date of the Winter solstice. Early Christians considered this a likely date for their saviour's nativity, as it was commonly held that the world was created on Spring equinox (thought to fall on 25 March at the time), and that Christ had been conceived on that date, being born 9 months later on Winter solstice.
A Swedish source states that the date of (Winter Solstice, St. Lucia, Lucinatta, Lucia-day, Lussi-mass ...) i.e. 13 December, predates the Gregorian which implies that "Lucia's Day" was 13 Dec in the Julian Calendar, which is equal to 21 December in the Gregorian, i.e. now. Same source states use of the name "Little Yule" for the day, that it was among the most important days of the year, that it marked the start of Christmas month, and that with the move to the Gregorian calendar (in Sweden 1753) the date (not the celebration) "completely lost its appropriateness/significance".
Saint Lucy's Day