Going back to the earlier posts of this thread, I have never come across Mary Hodgson being referred to as Dorothy. Perhaps it was a name she adopted for some reason? A search of official records, other than the British censuses dealt with by ecb, reveals:
Mary's birth was registered in the Kendal Registration District (which included Kirkby Lonsdale, her place of birth) sometime in the December quarter of 1876 (ie she was born in either October, November or December, 1876). Her name was recorded as simply Mary Hodgson, and therefore her birth certificate should show the exact same name. Anyone wishing to obtain a copy of her birth certificate will need also to quote Volume 10b, page 687.
If Mary was christened it is possible that an additional first name may have been recorded on that occasion. This could be checked locally, starting at the churches in Kirkby Lonsdale. Her christening is not shown in the International Genealogical Index (IGI).
Tom Hodgson was Mary's younger brother, yes, but she also had three older brothers and two older sisters.
There were no Dorothy Hodgson, or Dorothy Mary Hodgson, births registered around that time in that part of the country. A Dorothy Mary Hodgson was born on 2nd September 1884 in Sydenham, and her nickname was Datta. By 1929 she was living in India, and there seems no obvious connection with, or interest in the Davies family or Barrie. Anyone wanting to rule this person out should start by reading about her on the Web (just Google her full name).
I think the most likely explanation here is that the book belonged to a Dorothy Mary Hodgson who was a relation of Mary's. Neither of Mary's sisters had that name, but perhaps an aunt, cousin or niece did.