There are competing theories for the origins of the term, none of which is definitive. The Oxford English Dictionary gives the earliest attestations from Britain in the 1830’s, defining it as “the first week-day after Christmas-day, observed as a holiday on which post-men, errand-boys, and servants of various kinds expect to receive a Christmas-box”.
The term “Christmas-box” dates back to the 17th century, and among other things meant:
A present or gratuity given at Christmas: in Great Britain, usually confined to gratuities given to those who are supposed to have a vague claim upon the donor for services rendered to him as one of the general public by whom they are employed and paid, or as a customer of their legal employer; the undefined theory being that as they have done offices for this person, for which he has not directly paid them, some direct acknowledgement is becoming at Christmas.
In Britain, it was a custom for tradespeople to collect “Christmas boxes” of money or presents on the first weekday after Christmas as thanks for good service throughout the year. This is mentioned in Samuel Pepys’ diary entry for 19 December 1663. This custom is linked to an older British tradition: since they would have to wait on their masters on Christmas Day, the servants of the wealthy were allowed the next day to visit their families. The employers would give each servant a box to take home containing gifts, bonuses and sometimes leftover food.
The European tradition, which has long included giving money and other gifts to those who were needy and in service positions, has been dated to the Middle Ages, but the exact origin is unknown. It is believed to be in reference to the Alms Box placed in areas of worship to collect donations to the poor. Also, it may come from a custom in the late Roman/early Christian era, wherein metal boxes placed outside churches were used to collect special offerings tied to the Feast of Saint Stephen, which in the Western Church falls on the same day as Boxing Day.
Source: Wikipedia