Definition: 1) A diminutive of the given name Laurence, from the Roman cognomen Laurentius, meaning "of Laurentum" - a city in ancient Italy. 2) Possibly derived from the Old English 'hlaw' or 'hyll,' meaning small hill or burial mound; which became "low" in the south, but "law" in the north.
Many surnames are based on the place where the original holders of such names actually lived. Thus the earliest people called 'Marsh' most probably lived in a boggy area, whilst someone called 'Lea/Lee' lived by a meadow and someone else called 'Wood' lived by or in a forest. On that basis, 'Law' was most probably connected with someone who lived on a hill...just as people actually called 'Hill' probably were!
Dear Rod, as a matter of interest, 'Moon' might be a variant of 'Mohun'. Another possibility is that it may relate to place after all...ie from the island of Anglesey, originally called 'Mona'. That name may have come from the Welsh word 'mwyn', which relates to mines or quarries. Other conceivable meanings from Welsh are 'alone' or 'affable'. So, the earliest Mr Moon may well have been a pleasant Welsh miner who was a bit of a loner!