By now you've discovered much more on your Hadden ancestors, and with William J. Hadden and Mary E. Goodman's 1893 New Jersey marriage record, you found that William's father was "James" Hadden and his mother's maiden name was MOORE. This leads to a marriage in County Tyrone, Ireland.
Lower Clonaneese Presbyterian Church records, Co. Tyrone, Ireland - MARRIAGES
18 Feb 1870 - James Hadden & Mary Ann Moore
James Hadden, 25, bachelor, farmer
Father: John Hadden, farmer
Mary Ann Moore, 22, spinster
Father: James Moore, farmer
Witnesses: Thomas Coote, Margaret Hadden
Lower Clonaneese Presbyterian Church baptism records show children of this couple: William (born 1870), Isabella (1871), Eliza Jane (1873) and Ellen (1876). John (born 1879, according to civil registration records) is probably in the church register, too, but there are damaged areas of the original pages and some 1879 entries are obliterated.
You can download & view scans of the original register pages (15 MB) at www.clonaneese.org.uk/baptisms1840-1912.htm
In 1880, James and Mary A. Hadden arrived at the port of New York on the ship "City of Montreal" with two of their children.
James Hodden (Hadden) 32 - country to which this person belongs: England - destination: U.S. - passenger: steerage
Mary A. - wife - 28
Isabille - child - 6
Ellen - inf. - 1
I haven't yet seen a ship record for William James and Eliza Jane ("Jennie") Hadden, but they certainly came to New York, too. They were probably with their parents (but not necessarily, even at that young an age); the shipping clerk may have just missed them in his list. Also they may have been brought over a few years later with aunts/uncles, cousins, or grandparents. This wasn't an uncommon practice, so that an entire family wouldn't be lost with a sinking ship in the trans-Atlantic passage.
Eliza/Elizabeth Jane "Jennie" Hadden married Henry HENNINGS in 1892 in Manhattan, and (as you found) William James Hadden married Mary E. Goodman in Camden, New Jersey in 1893. I haven't traced Isabella, Ellen or John.
James Alexander HADDEN (William James' father) was the son of John Hadden of Farriter, a townland in Killeeshil parish, County Tyrone. John Hadden was born about 1810, judging by his death certificate.
Death certificate from GRO, Dublin:
Date & place of death: Twenty ninth October 1870, Farriter
John Hadden, male, widower, 60 years, Farmer
Cause & duration: Disease of heart, 5 or 6 years, uncertified, no medical attendant
John Hadden, Jr., present at death
Registered: Fourth of November 1870
John Hadden of Farriter's identified children included:
1. Mary Jane who married Samuel Cherry
2. James Alexander Hadden who married Mary Ann Moore and emigrated to the United States
3. John Hadden (Jr.) who married Ellen Beatty
4. Margaret Hadden who married William James McLaurin
5. Sarah Hadden who married Thomas Coote
6. Anne Hadden who married James McIlmurray
John's father was probably the James Hadden counted at Farriter in Ireland's Tithe Applotments, 1829.
I have more information on some of John's children and will be glad to share research notes. The Haddens of County Tyrone have been a fascinating family to research. My own Hadden ancestors lived in and around Ballygawley and Aughnacloy, in Carnteel and Aghaloo parishes.