There have been people living around the Norwich area since at least Roman times, and in fact there was an important Roman city in the area, but the Roman town isn't the origin of the name. In Anglo-Saxon times, around the 5th to 7th century, there
was another small settlement on the river Wensum, which became known as Northwic or
North Wic. 'Wic' is an Anglo-Saxon word meaning port, and it is thought that
there may at one time have been two settlements, one south of the Wensum and
one north, and the name of Norwich derives from the northern one. It certainly
became a river port quite early, and steadily grew until in the Middle Ages it
was one of the most important cities in England.

THe county of Norfolk, of which Norwich is the capital, is also named for its
northern location - in this case as opposed to Suffolk, its neighbour to the
south.