Thanks very much RobbieHL for stopping me "barking up the wrong tree" for probably a long time! Though my faith that we had the right Mary was beginning to be shaken as the "Baillieston Mums" got too scarce and very old! And forgive me for attempting to assign to your ancestry a lady of such unfortunate circumstances.
I now have to regroup...though most will be somewhat jaded with this search by now!
Assuming that she told the truth to the Inspector, here is what I know about our Mary:
She is named Mary Gilchrist (or Ghildchrist), a former domestic servant and mother of an illegitimate son Henry (Hendrey) who was born at 5.30 am on Wednesday,15 May 1878 at 3 Taylor’s Close in the Middle Parish of Greenock, Renfrew County, Scotland.
In the 1881 Census of Scotland, Mary Gilchrist is listed as being in the Greenock House of Refuge, 9 Upper Ingleston, Back of Land, Greenock; born in Holytown, aged 25, a washer. But I also have 1881 Census Scotland ,Renfrewshire Greenock East ED24/ Reg # 564/2 House of Refuge GILCHRIST Mary 32 Washer born Hollytown! Which one is right or are there two? The first matches the age of our Mary.
In the 1891 Census Scotland Lanarkshire Barony Reg # 622/2 Roll CSSCT1891/222
Convent of the Good Shepherd, Refuge or Home of the Good Shepherd and Reformatory School for Girls
Dalbeth London Road - Glasgow a GILCHRIST Mary aged 40 is listed as Inmate of the Refuge born Bothwell Lanarkshire. This one matches the age of the second Mary above and not ours.
In the 1901 Census Scotland Lanarkshire Govan Reg # 646/3 Roll CSSCT1901/332 Royal Asylum Gartnavel Glasgow, a GILCHRIST Mary 48 born Lanark, is a Patient but is not a good age match for any of the above.
From the Quarrier Archive Records:
On 30 January 1882 Mary was 27 years old and had been in trouble for 5 years.
Henry who would be 4 on the 15 May 1882 had stayed for a short time with Mary's mother who lived in Rankine's Land (Raukine's Laud or Lane), Baillieston
Mary's father died many years ago.
Mary had moved with Henry to Greenock to stay with Mrs Fullarton, 6 Darymple Street, and three other girls in a house of dubious repute.
Mary was being abused allegedly by Mrs Fullarton and had sheltered in a House of Refuge ( presumably the Greenock House of Refuge in 1881 as per the census), but was induced to return to her sad life.
On 21 Nov 1891 a letter to Quarriers from the Matron of Greenock House of Refuge asking about Henry states that the mother ( Mary) is there at present and that some time ago she was in Dalbeth Refuge.
In June of 1914 June, Mc(

) Berry, of Mayfield High Township Nr Greenock writes to Quarriers about Henry and his mother (Mary) is said to be in Dalbeth ( Refuge).
There is a Langloan Rankines Land, Old Monkland Middle District Lanarkshire; also a Rankine's Land address in Stirlingshire.
The 1881 Census of Scotland has Elizabeth Fullarton living at 6 Dalrymple St, Greenock, a widow, aged 52, employed as a 'broker'. In the same house are a daughter, niece and two female servants.
In theory, Mary Gilchrist was born circa 1855 to a Gilchrist family living in , or near Baillieston. Her mother was still living there in 1882 and so is likely to have been there in the 1881 census. There is also an outside chance that Mary's father's name was named Henry given Scottish naming tradition. So a marriage of a Henry Gilchrist say 1840-1855 and a birth of Henry Gilchrist say 1820 in the same vicinity might be interesting. Mary likely died after 1914 after living in and out of several Refuges including Dalbeth.
So that is all we know so far!
RobbieHL, does any of this ring any old Gilchrist bells from your research?
I think some of the Archive material from the Refuges if it still exists could help. She could well be buried at Dalbeth ( St Peter's new cemetary?).
Best wishes, Lumber-Jack