This comes from an older expression 'fourteen nights', which gradually got worn down to 'fortnight'. Until about the end of the 18th century there was a similar expression 'a sennight'. This came from the phrase 'seven nights' or in other words a week.

A lot of time expressions used to be different in the past. For example, the word 'weekend' is very modern - it wasn't really used until the 20th century, and was considered quite vulgar until well into the middle of the century. The upper classes preferred the expression 'Saturday to Monday'. This was probably because the whole point of a weekend is a break from work, and until relatively recently, people typically either worked six days a week (with Sunday often called the 'Sabbath' and seen as a holy day of rest) or they didn't work at all if they were rich enough (or not at 9 to 5 jobs anyway).