The name is the name of the land the family settled on, it does not indicate where they came from. Although the name is Anglo-Saxon, there is no reason to suppose that the family is. The family could be Celtic, Norman, Anglo-Saxon, or even Viking! There are clear signs of all of these peoples settling in the surrounding area.
When people first started to write the name down, there was no particular way to spell it, so many different spellings were used. Records show that at least 29 different spellings have been used in the past:
Farsacharley Fasackerlegh Fatzackerley
Fazacherly Fazackarley Fazackerleigh
Fazackerley Fazackerly Fazacrelegh
Fazakarley Fazakarly Fazakerley
Fazakerly Fizacarley Fitzackerley
Fitzakerley Fizakerley Phasakyrlee
Phaseakerley Phazakerley Phizacarley
Pheazackerley Phezackerley Phezakerley
Phizackerley Phizacklea Phizaclea
As far as I am aware, the names in bold are the only ones still in use today.
The earliest writing used the spelling "Fasackerlegh" and "Fazackerleigh". The most common is "Fazakerley", probably due to the fact that it became the accepted spelling for the town. People used the towns name as reference. As we move further away from the town of Fazakerley, so the spelling starts to change!
Different spellings are regional. This is because families used to rely on officials, who were creating the records , mainly priests, to know how to spell the name, and those officials used their own particular spelling to apply to within their jurisdiction or Parish. Certainly some of the "ph" spellings seem to have been restricted to individual towns. My particular spelling, "Fazackerley" originated outside of Liverpool, perhaps from Wigan, and it became particularly popular in Preston.