A quick Internet search for the meaning of the name Levers does not yield any actual concrete answers. The name Levers and its fellow Leivers are often given different origins. Yet, the Greasley family of Levers and Leivers are all one.
Clearly there are different views the origins and meanings on all surnames and the various pockets of the the names will probably all have diverse origins.
Out of interest, here are some meanings of the name Levers found on the Internet.
From the Ancestry.co.uk website – Levers name meaning
English: patronymic from Lever 3.
Source: Dictionary of American Family Names ©2013, Oxford University Press
2) From Forebears.io website
Levers surname definition – this surname is derived from a geographical locality “of Lever”. Great Lever is a township in the parish of Middleton, Lancashire and Little Lever is a chapelry in the parish of Bolton, Lancashire.
Source: A Dictionary of English and Welsh Surnames (1896) by Charles ~Wareing Endell Bardsley
A personal name hence the genitive Levers, the patronymical Leverson and the local Leverton, but it is also local, there being three places named Lever in Lancashire.
Source:Patronimica Brittanica (1860) Mark Anthony Lower
3 )From thehouseofnames.com
The Levers name originates from the places in Lancashire of Little Lever and Great Lever.
However
When researching the meaning and origins of the name Leivers, the addition of the extra letter “i” makes a huge difference. From the Internet Surname Database https://www.surnamedb.com/Surname/Leivers this suggests that the name is French in origin, possibly from the Old French “Lievre” introduced to Britain by the Huguenot Refugees. Also may be from the patronymic form of the medieval French “Levre” meaning a fast runner or messenger, from the Olde English “Leveret”, a young hare.
Variants
the website your-family-history.com gives no sources for its suggesting that the following names are all variants.
Lavers Levers Leavers Leivers Olivers Livers Leevers Lavars Lovers