It was originally called Pigh Hill, and it comes from a Middle English word meaning a small enclosure or croft.
Info from Sheffield Museums site.
Heths
Hi Heather
Can I just add, without distracting from your excellent work, that there is a freely available book available on the Internet Archive which has other old terms for Sheffield (link below), and may be useful for similar queries:
A GLOSSARY OF WORDS USED IN THE NEIGHBOURHOOD OF SHEFFIELD
Including
A SELECTION OF LOCAL NAMES, AND SOME NOTICES OF
FOLK-LORE, GAMES, AND CUSTOMS
GATHERED AND EDITED
BY SIDNEY OLDALL ADDY, M.A.
1888
Extraxt:
PYE-BANK, sb. the name of a street in Sheffield.
It occurs in Gosling's Map, 1736. There was formerly a family called Pye in Sheffield. ' Imprimis Pay banke field (arable) next unto the two last pieces east and north and next unto Payhame banke in part west and containing 9a. oor. ' --- Harrison.
A meadow called 'Pigh hill,' ibid. Pye Banke,' 'Pye Banke hill,' ibid. Pye greave, a field in Ecclesall, anno 1807. See PIGH HILL.
Now that is an interesting link! I know what I’ll be reading this week!
I did wonder if there might have been a link with a family called Pye, so I had a look at the reminiscences of Twiss/Wragg/Leader.
There was a very brief mention of Pye Bank, and of a Dr. Pye Smith, but they didn’t make any connection between the two. They only said that the Doctor had been born on Snig Hill.