Sorry for the error in previous message (I've corrected that in red print). Yes, Fanny Wilson was the person who registered James' birth in July 1884. I haven't yet found a son George for George Wilson & Fanny Greenaway but there certainly isn't a marriage between a George Wilson & Eliza Greenaway in Ireland.
I think that when Fanny Wilson registered James' death, whether by design or by accident, she gave as the father's name that of her husband, George Wilson, and for the mother her daughter Eliza with her own maiden name. By the time of the registration Eliza was probably already quite ill and it would be left to his grandparents to look after the baby. It's possible, although impossible to prove, that Eliza also suffered from tuberculosis and pregnancy would have weakened her further.
Back at the start Laura mentioned that this James Wilson is listed as Thomas Wilson's son but not mentioned in wife's obituary. Another bit of the story was that James was sent to America when an orphan. Fanny Wilson died when James was 4 months old so perhaps one of Eliza's sisters looked after the baby. George Wilson died in 1889 and it seems James went to America sometime after that date. It's entirely possible that James was never told much about his mother and indeed his father.
When I looked through PRONI's birth registration index, which it doesn't look like Laura did, I found the daughters I mentioned in reply #7 but continued from 1871 for quite a few years in case Fanny had a much younger child. I was surprised to find a Wilson-Greenaway birth for a James Wilson in 1884 (Fanny was 57 at the time) 13 years after last child and thought that George & Fanny were the parents. It was only when I looked at the actual certificate, and Eliza's death certificate, that things started to fall into place.
It's possible a baptismal record for infant James Wilson will list Eliza as his mother with no father's name listed.
One thing I don't understand is how Laura missed seeing James' birth in the actual certificates on Irish Genealogy since she was looking for a James Wilson born 1884 in that area