My great-grandmother, a widow at the time, had two 2-year-old "lodgers" with her in 1911; they don't seem to be related in any way to her and I can't trace their parents.
My father said that the family used to take in orphans and get money for looking after them (I don't know who from though).
So the 11 year old "lodger" may have actually been in the position of what today we would call a foster child.