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“The Aran Islands.”  by J.am.Synge.

That would be none other than John Millington Synge, as in the writer of "The Playboy of the Western World".

I cannot see any newspaper references to it.  But the first thought that comes to mind is that it's the altitude above which briar ceases to flourish in hilly / mountainous areas... a bit like the notion of the tree line.

Added:  It pretty well implies same here: "Many people from the surrounding area fled to those mountains so that they could live above the "briar line," the briar, it was believed, having been the main carrier of potato blight spores."

https://www.sligoheritage.com/ArchBlack47Keadue.htm

I can see a transcription of a baptism at St. Mary's Belfast on 20 May 1883 for a Mary Jane "Marran" for parents Robert "Marran" and Mary "Monaghan" of 5 Chapel Lane (there's that address again), the sponsor was Mary McCormick, I wonder if this is a mistranscription of Robert Morrow and Mary Malaghan?  Unfortunately, not available to view on the NLI RC parish records website. Can't see a civil registration either. I was hoping to track Robert and Mary forward, for example, to see them in the 1901 Ireland Census (Mary's birth place would be nice to see).

Anyhow, here are some listings for Chapel Lane in the Belfast Street Directory:

1880: 5 Malaghan, Mrs., green grocer
1884: 5 Monaghan, Susan

I think I’m with you, Gaffy. It seems a bit complex, doesn’t it.
Here is the parish record of the marriage of Catherine Mallaghan and James Owens. They have been recorded as James Jones and Catherine Mallon but all the other details are correct.
27th January 1874

https://registers.nli.ie/registers/vtls000633206?locale=en#page/43/mode/1up

This Catherine has a mother Frances. The witness, John Mallaghan has the address Marquis Street.
This does not answer the puzzle about Susan Mallaghan.
Perhaps the two Catherines were cousins and their two fathers brothers.

Great find, I would never thought of looking under Mallon. It must be something like that, if not brothers, maybe cousins. 

OP, the ages and dates indicate that we may already be too far back to find any useful records, this may prove to be a brick wall. I can't see any marriage for Patrick and Susan, nor baptisms for daughters Catherine and Mary.  A couple of observations: (1) I noticed other Belfast Mal(l)aghans/Malligans on the go who were all sons of Henry (Edward, Thomas, Peter, Patrick) and who were all dealers or auctioneers or brokers, I wouldn't be surprised if related back in time; (2) a quick check of older ones with this surname in the 1901 Ireland Census shows a cluster in the Ballymena area and others pointing back to counties Tyrone and Londonderry, so Patrick and ancestors might have been blow ins to Belfast.

Gaffy, check out the marriage record for Catherine in the opening post.

Sorry, I don't get it. Who is Thomas, how does he fit in to this?