I'm suffering with a drawn-out cold and not firing on all cylinders, so I'd be glad of any help in trying to figure out this puzzle.
In 1871 Edmond L Rain aged 4 is in Garrigill, Alston, listed as the son of widow Ann Rain - RG10/5199/51/12
Ann Rain 55 Widow
Johnny Rain 23 son
Henry Rain 16 son
Edmond L Rain son 4
By 1881 he is in Teesdale with another of Ann's sons, Robert - RG11/4936/97/1
Robert Raine 38 b Alston, Cumberland
Mary 45 wife b Alston
Edmund 14 son b Alston
Robert married Mary Carr in 1870.
He is still with Robert in 1891 & 1901 (latterly with his wife and young children), and always described as Robert's son.
I can't see any Edmund/Edmond Rain/e born in 1866/7, but Edmund Larol DODD was baptised at Garrigill on 13 Sep 1866, the illegitimate son of Elizabeth Dodd.
The two main candidates for Elizabeth Dodd are the one born in 1842 mmn Bradwell (John Dodd m Mary Bradwell in 1821), or born 1847, mmn Spark (William Dodd m Elizabeth Spark in 1837).
In 1871 I think the older Elizabeth Dodd is living alone in Alston (RG10/5197/14/22), but I'm not sure what happened to her after that.
The younger Elizabeth Dodd married William Hunt in 1869, Carlisle RD, and is with him in Teesdale in 1871 (RG10/4947/40/32). An Elizabeth Hunt aged 28 died in Teesdale RD in 1876.
In the absence of any other explanation I've assumed that Robert was (or believed he was) Edmund's biological father, although his naming 'Edmund Larol' seems very specific and doesn't fit anywhere into the Rain/e family, as far as I can see.
I've no idea which Elizabeth might be the mother, although the Teesdale connection may be relevant.
My fuzzy brain thinks this is probably unsolvable, but I'm open to suggestions and/or any records I may have missed.