The names according to the Concise Oxford Dictionary of Place Names, the j31 area villages and hamlets (for such they were, when they were founded, even cities like Sheffield). The designation DB = Domesday Book of circa 1086. The suffix word Tun appears quite a lot, and is Saxon or Old English (more than 1000 years old) for fence or enclosure. But often morphs into meaning village or town in later centuries. So Beighton could be loosley translated as Beck Town. Beck is from Old Norse bekkr indicating Viking roots. Our place names are often ancient.
Part of England's DNA...
Anston - In the Domesday Book (DB) 1189 CE. Old English Anna Stan or single stone. Or maybe even Anna's Stone. Either way, Anston's Stones seem to have stamped their identity on the area. Limestone from Anston was used to build the Houses of Parliament.
Aston - DB Es Tun - East Enclosure. There are dozens of Aston's in England. I assume the many Norton's, Weston's and Sutton's are derived from north, west and south tuns.
Aughton - DB Actone - Oak Tun (enclosure, or even town)
Beighton - DB Bectun - Tun by the brook. Derived from Middle English 'bece' or vally of a stream. I suspect it's where we get the modern word Beck for brook/stream.
Brampton - DB Brantune - Old English Bromtun - Tun Where Broom Grew
Dinnington - DB Dunnitone - Tun of Dynne's People.
Harthill - DB Hertil - Hill Frequented by Stags.
Killamarsh - DB Chinewoldemaresc - Cynewald's marsh.
Kiveton - DB Cieutone - Old English for A Hill?
Laughton - DB Lachestone - From Old English Leactun or kitchen garden of leeks.
(Laughton) en le Morthen - maybe from Mordhingg or The Moor District
Shireoaks - appeared in 1286 as Scirakes or the Shire Meeting Place.
Swallownest - possibly named after a farm of the Swallow familiy. Part of Aston cum Aughton.
Treeton - DB Trectone - Tun by the Trees.
Thorpe Salvin - DB Torp (Old Danish for farm) - Salvin is a later 1300ish Old French nickname meaning Savage (salvagin).
Throapham - not given - if you know, tell me...
Turnerwood -I'm guessing a fairly modern name from Turner's Wood?
Thurcroft - Thori's Croft - Croft is Old English for enclosed land used for crops or pasture. Thurcroft didn't exist as a town until the pit was sunk in the 20th century.
Todwick - DB Tatewick - Tata's Wic; a wic being from the latin victus or dwelling place/village
Ulley - DB Ollei - Old English for Wulf Leah or wolf wood. The W may have been lost due to Norse influence. An interesting comparison with Woodall.
Wales - DB Walis - almost the same name 1000 years later. Old English Walis for 'The Welsh'.
Woodall - not given in the Oxford book, but Woodale in North Yorks means 'Wolves Valley'.
Woodsetts - in 1324 recorded as Wodesete or Fold in a Wood.