Hi Judy,
I'm not sure I can help with tracing the family, but I can certainly supply you with information about the area, as I live in Prestwich! Incidentally, your N.B. is a little wide of the mark - these days Prestwich is part of the Bury local authority area, which borders Manchester on its (Bury's) southern edge.
Using your info I spotted a birth record for a Rebecca Martin, birth registered in the June 1859 quarter in the Chorlton registration district. Chorlton covered part of South Manchester, whereas of course Strangeways was (and is) north of Manchester City Centre. No idea whether this is 'your' Rebecca - we'd need the actual birth certificate to check. Anyway let me know if I can be of any more help.
Kind regards, Philip