RootsChat.Com

Note, to avoid duplicate work, that the same topic has been started under "The Common Room":

https://www.rootschat.com/forum/index.php?topic=893026.msg7672065#msg7672065

On top of the evidence indicating the connection between William King Lynar, Eliza Glover, and the children Alfred, Olivia Jane and Jonathon Joseph, here are another couple of references that corroborate the sibling relationship between the two surviving children Alfred and Olivia Jane and suggest continuing contact between the respective families years later. The first reference is the death notice for Alfred in the Belfast Telegraph of 13 August 1923: 

LYNER - August 12, 1923, at his residence, 26 Flora Street, Alfred, the dearly loved and loving husband of Martha Lyner.  His remains will be removed from the above address for interment in Dundonald Cemetery, on Tuesday, the 14th inst., at 2.30 p.m. "Safe in the arms of Jesus." Deeply regretted by his sorrowing Wife and Sons, Daughters-in-law, and Grandchildren, also Sister and Brother-in-law and Nephew.  MARTHA LYNER.

So as well as Alfred's own family, the only other relations getting a mention were a "Sister and Brother-in-law and Nephew", a description that is consistent with the only other relatives we have evidence of in this story so far, namely Olivia and Edmund Costley whose only child was a son - William John - and therefore a nephew to Alfred. It would have been nice if the Costleys had actually been named, still, from the way its written I'm pretty confident that it's them.

The second reference relates to Alfred's son Edward.  The Belfast Newsletter of 25 November 1914 reported the death of Edward Lynar from his war wounds and said that he "was a member of Roe Memorial Total Abstinence L.O.L. 938, Ballymacarrett". Working back from that, on 25 February 1911, the Belfast Weekly Telegraph reported on a social evening held by the Roe Memorial L.O.L. No. 938, at which one of the ladies mentioned as presiding at the tea tables was a Mrs. Costley.  There was also mention of a Br. Edward Lyner contributing to the programme (of entertainment) and a Br. Edmund Costley being the Secretary of the Lodge.  The same newspaper of 16 February 1911, covering a reunion of members of the Roe Memorial L.O.L. 938, named the secretary as Br. Edmund J. Costley and mentioned that Br. Edward Lyner provided a recitation as part of the musical programme.

Now if these were run-of-the-mill names, it would be no great deal, but given how VERY rare - if not unique - these specific forename/surname combinations were (as can be verified by the 1911 Ireland census), I think it is way beyond mere co-incidence that they should appear together in the one context and to me it indicates the probability that Olivia Jane Costley's family and Alfred Lynar's family knew each other and occasionally interacted/associated.

You'll find several "Lynars" and "Lyners" in the Belfast City Council burials website, I've only glanced at them, but because we now know that the surname is rare and suspect that all of the ones found so far in Belfast and South East Antrim are a cluster connected through William K. Lynar and Elizabeth Glover, I would imagine that they are all relevant:

https://www.belfastcity.gov.uk/births-deaths-and-ceremonies/burial-records/search-for-a-burial-record

Olivia Jane and the mother Elizabeth are in a Costley grave on the same website.

... I am also supprised to read in gaffa's last post, a newspaper extract dated 14 August 1869, stating that Eliza Glover had taken her 3 children to Whitehead and left them with Mr Eslar a Warden. Was he the C of I church warden?

This may have been William or David Esler, two brothers who were prominent in the local (Whitehead) community, as they owned between them a farm of 90+ acres in the immediate vicinity. They would have been aged in their late 60s back then.  If not them, then possibly a son like Patrick Esler. 

In the context of the newspaper item and poor relief, this reference is more likely to mean that Esler was a warden for his dispensary district, meaning that he could issue tickets for relief (medical or other) within that district. So he would have been a local point of contact in the Whitehead area for accessing such assistance and all related matters.