Tatiehead,
I looked in the 1901 census and there was only 1 Brawley in Co Derry. This was her:
http://www.census.nationalarchives.ie/pages/1901/Londonderry/Dungiven/Dungiven/1524346/
There were 259 Brolley/Brolly though. That looks to be an alternative spelling of the name:
The 1831 census has 7 Brawley housesholds in the county. (All RC). None in Glack, and none headed by a William. (James x 2, John, Edward, Pat x 2 & Giles).
I did find a James Brolleys in Glack.
http://www.census.nationalarchives.ie/pages/1831/Londonderry/Kennaught/Tamlaght_Finlagan/Glack/49/ 4 males and 1 female in that family.
There was also a Daniel living nearby with 1 male and 3 females. He’s listed as Brotley on the transcript but it looks like Brolley to me on the original.
http://www.census.nationalarchives.ie/pages/1831/Londonderry/Kennaught/Tamlaght_Finlagan/Glack/51/
Unfortunately the RC records for Glack (Drumachose parish) don’t start till 1855 so if this is where your family originated it may be hard to prove.
I searched for deaths in the area (Limavady civil registration district) 1864 – 1901. I found 91 Brolly. (Nil Brawley). You might want to go through them to see if any are from Glack.
https://civilrecords.irishgenealogy.ie/churchrecords/civil-search.jsp
In Griffiths Valuation (1858) there were 2 Brolly households in Glack. Patrick had plot 1 which was a 13 acre farm. He was subletting a farm labourers cottage to a Margaret Brolly (probably a relation and probably a widow). Also on plot 7 there was another Margaret Brolly, again in a labourers cottage.
http://www.askaboutireland.ie/griffith-valuation/index.xml?action=nameS…