Nobody knows for sure, but the general consensus is that the word is derived from lamp-post. Over thousands of years, the system of messengers running here, there and everywhere with written messages developed into the post service. It really took off in England when King Henry VIII appointed a master of the Post in 1516 to maintain a regular postal service along the main roads radiating from London. There were places where a box was attached to a lamp-post and a post-master (usually an innkeeper) was responsible for passing a piece of mail onto the next lamp-post. A stagecoach was used for this purpose and the mail went down the line until it reached the destination. Later, the same practice was adopted by the USA.
These staging posts also served as stop-off points for mail coaches, where the coachman could rest or change his horses. This is how we got the terms ‘post-box’, ‘postage’, ‘post office’ and ‘postman’.