Hello again,
I have had a look at the 1901 census and can not find a Lillian Armstrong, but I have found this family which includes some of the names you mentioned.
1901 Census - RG13/4779 folio 135, page 15
3 Factory Yard, St Andrews, Newcastle, (district 5)
Mary Armstrong, head widow 50 - Newcastle
George T. Armstrong son single 27 French Polisher Newcastle
Annie Armstrong dau. single 23 - Newcastle
Joseph B. Armstrong son single 17 Moulder Apprentice Newcastle
Florence L. Armstrong dau. single 13 - Newcastle
William J. Armstrong son single 11 - Newcastle
Ethel M. Armstrong dau. single 9 - Newcastle
Edith M. Armstrong dau. single 7 - Newcastle
I was wondering if the Florence L. Armstrong might be Florence Lillian?
10 years earlier in 1891 the above family were living at 4 Camden Street, Newcastle.
1891 census - RG12/4212 Folio 15 pages 22 & 23 All Saints (District 11)
George Armstrong Head Married 40 Quay Labourer Benwell
Mary Armstrong Wife Married 40 - Newcastle
Elizabeth Armstrong Dau. Single 22 - Gateshead
George T. Armstrong Son Single 17 French Polisher App. -
Isabella Armstrong Dau. Single 16 - Newcastle
Annie Armstrong Dau. Single 13 - Newcastle
Agnes A. Armstrong Dau. Single 10 - Newcastle
Joseph B. Armstrong Son Single 9 - Newcastle
Florence L. Armstrong Dau. Single 3 - Newcastle
William J. Armstrong Son Single 1 - Newcastle
It also states the son George T. was deaf & dumb from childhood.
1881 Census - RG11/5059 Folio 73 page 3 - St. Andrews Newcastle
4 St. Andrews Courts (next to Pilgrim Street)
George Armstrong Head Married 36 Painter
? Benwell
Mary Armstrong Wife Married 31 - Newcastle
Mary Armstrong Dau. Single 14 Scholar Gateshead
Elizabeth Armstrong Dau. Single 12 Scholar Gateshead
George Armstrong Son Single 8 - Newcastle
Isabella Armstrong Dau. Single 6 Scholar Newcastle
Annie Armstrong Dau. Single 4 Scholar Newcastle
Agnes Angus Armstrong Dau. Single 2 - Newcastle
You would really need to check these details for yourself, because we all see something different in the squiggles!!!!!
I have found a birth record for a Florence Armstrong September quarter 1887 Newcastle 10b 62, but there is no hint of a second name of Lillian.
Good Luck with your search and I hope this does not send you along the wrong road.
Best Wishes, Patsy