We know that William had a sister named Elizabeth & it was tradition to name children after their parents, just as William was named after his father. We also know that Clones is only 15 miles from FivemileTown.
1.) If I can find a surviving Census record for 1821, we might identify other members of the family, which would hopefully include William's brother, John Armstrong.
2.) If we could find William Armstrong Sr.'s father's Will, he should name all the members of the family and hopefully mention Mill house by name.
3.) If we could identify John Armstrong's burial site, his headstone may include the name of his parents as well as his wife & other members of the family.
A discovering of any of these sources would help to confirm or reject my current theory.
We should have some extra time later this week over Thanksgiving to do some more digging! HAPPY THANKSGIVING WEEK!