When the noun mission was borrowed into the English language in a 1513 theological text, it referred exclusively to God sending Jesus onto Earth. Around 1598, a new definition emerged, describing Jesuits who were sent to Europe, and a few decades later it was broadened to anyone being sent anywhere with a purpose. The word always had to do with sending, though, and that's because it derives from Latin missionem, meaning "sending". That's the fourth principal part of the verb for "send", mittere (also the source of words like emit, manumission, transmit, submit, admit, compromise, and missile, among others), and that ultimately derives from the Proto-Indo-European reconstruction meyth, meaning "to exchange". Usage of the word mission over time has been pretty constant since the 1800s.
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.