Originally, the word average referred to "financial loss from goods being damaged in transit". This might seem very distant from the modern definition, but the connection is that anyone who invested in a ship with lost/damaged goods (or their insurers) had to turn in the average value of the damage. So, for a while, "average" meant "equal sharing of loss", and by 1755 the word took on its current mathematical connotation. Average was borrowed in the late 1400s from French avarie, which referred to ship damage in general. Beyond that, there are many possible explanations for the etymology; in fact, it's one of the most investigated origins, to no avail. It might be from Arabic awar, meaning "defect", Latin habere, meaning "to have", or various other linguistic permutations.
Adam Aleksic, a rising sophomore studying linguistics and government at Harvard University, has been described as the internet's sixth most famous etymologist. He also has disturbing interests in politics, vexillology, geography, board games, conlanging, and law.