According to David Linstead Heraldry - The surname "Clarence" was derived as follows:-
CLARENCE is a form of the name of Clare, a Norman-French place-name for one from St Clare or St. Clair in France.\
Lionel Plantagenet was titled Dux Clarensis by his father Edward III.
There is a Blazon of Arms for CLARENCE as well :
"Arms:- Per chevron gales and azure in chief two lions combattant or in base a fleur-de-lis of the last."
The adoption of second names in Europe took place from the 11th century onwards. Surnames generally fall into main categories:- Occupational names describe the person's profession. Locational names describe the person's dwelling or place of origin. Descriptive names distinguish the person by physical characteristic or mannerism. Patronyms are the adoption of the person's father's first name as a surname.