Hi Lu
Another curly one for our family is Eliza Cox (nee Fletcher) and her child by Patrick Boylan.
Her name on the birth is Caroline Boylan. Here is the link to where I got to but no takers
http://www.rootschat.com/forum/index.php/topic,446014.0.html
Hi Marj
Hmmm ... not surprised "there were no takers".
Even though you provided a link (to another thread), any potential helpers still need to do an enormous amount of reading, (and view links to other information already sourced), in order to get to grips with the "story". It all takes quite a bit of finding.

And maybe help would have been forthcoming if your initial query (here) had mentioned the child's name ... instead of "2nd daughter of Eliza Cox" .... and "this wee girl"
I did earlier look at the "COX- related" threads on the Australian Board (and need to do so again to refresh my memory).
My "take" on matters, is that it was the
elder daughter of Eliza COX who was taken into the custody of William and Christina FLETCHER (probably on a temporary basis ?). The infant BOYLAN child seems (for a time) to have been taken care of by Eliza's sister (a Mrs. BELL ... was it ?).
But aren't both of these girls noted as "living issue" on Eliza's death certificate ? I tend to think that Eliza, some time later, resumed responsibility for them ?
I would rule out the Caroline FLETCHER marriage 1905 to WYLLIE.
I have some information for "a Caroline" - whom I think is a much more likely "candidate"

- which I'll post shortly.
~ Lu