Formal adoption only started in Northern Ireland in 1927. Up till that time adoption/fostering was an informal arrangement with little or no paperwork. Can make people hard to trace.
Dungannon Workhouse does not appear to have any records pre 1899 so no clues there:
I had a look in the 1901 census around Newmills. There were 3 Browns living there but none was married. If adopted by that family Mary may have acquired the surname Brown. If just employed as a servant, then she’ll probably have kept the surname Tennyson.
Farlough townland where James Hammond farmed is part of Newmills, so perhaps the girls were all in the same general area? Here’s James Hammond in 1901:
There are some marriages in the general area for Eliza Janes and Marys (an example below):
Unfortunately Irish marriage certificates don’t contain the mother’s name and so there’s no easy way of knowing if it’s the right person.
Here’s the death of a Margaret Tennyson from Armagh, spinster, aged 53, who died in July 1901 in Omagh asylum. Don’t know if it’s your lady.
PRONI might have the patient records, which could tell you a bit more about the person.