In 1870-72, John Marius Wilson's Imperial Gazetteer of England and Wales described West Rainton like this:
RAINTON (West), a village, a township, and a chapelry, in Houghton-le-Spring parish, Durhamshire. The village stands ¾ of a mile N E of Leamside-Junction r.station, and 4¼ N E by N of Durham; and has a post-office under Fence-Houses. The township comprises 1, 788 acres. Real property, £5, 722; of which £2, 100 are in mines, and £810 in railway s. ...
Pop., 1, 447. Houses, 308. The manor belonged once to a convent at Durham, and belongs now to the Dean and Chapter of Durham. The chapelry is more extensive than the township; and was constituted in 1838, and re-constructed in 1866. Pop., about 3,000. The living is a rectory in the diocese of Durham. Value, £310.* Patron, the Bishop of Durham. A new church was built in 1864. There are a Wesleyan chapel and national schools.
GB Historical GIS / University of Portsmouth, History of West Rainton, in Durham and County Durham | Map and description, A Vision of Britain through Time.
Date accessed: 27th March 2017
Click here for more detailed advice on finding places within A Vision of Britain through Time , and maybe some references to other places called " West Rainton ".