Surnames can be classed into four main categories: Local surnames such as Bedford, Lincoln; Surnames of relationship, such as Johnson, Martin; Surnames of occupation, such as Miller, Major; and surnames derived from nicknames, such as Short, Little.Eurardus Ebrard--1086, Derbyshire
In the Southern half of England surnames adapted from first names were particularly popular, (surnames from places more popular in the North). The son would use his father's personal name as a by-name, this eventually evolving into a hereditary surname.
After the Norman conquest the personal name of Everard was introduced to Britain, being especially common in the South and East Anglia at around 1300AD. It was quite common for the land holding classes of Britain to adopt such continental names. Everett, Everatt, Evered, Everid, Everitt and Averitt are all surnames which evolved from the name Everard.
Everard has two likely origins. It may have developed from the Old English name Eoforheard, but it is more likely to have been adapted from the Old German name Eborhard. Both these variations have the same meaning however, with Eofor/Ebor meaning boar, and heard/hard meaning hard. Which means original Everard's had the characteristics of a hard boar, or were possibly hunters.
The adaptation of the surname over time can be seen when it occurs in various historical documents:
William Everard--1204, Bedfordshire
Geoffrey Everad--1300, Norfolk
So if you managed to trace your Everett lineage back to 1100-1300 it would probably end up with a man whose first name was something similar to Everard. It is also likely that they would have been in the Southern half of England--an area more affected by the Norman conquest.
And if Everett families had heraldry it would have looked like the coat of arms above.