I’ll give you what I’ve got but I wasn’t expecting anyone to look this up for me and to be honest I’m reasonably confident I haven’t left any stone unturned including quite similar searches to your suggestions but feel free to do some digging.
Now I only found out about William the other week when Ancestry added the 1875 NY State Census. Luckily for me my family lived at 96 Bergen Street from 1871 to 1875/6 which is Brooklyn. The family members I knew about were:
James O’Neil b1846 Ireland, occupation liquor dealer at 4 Water Street.
Hanna O’Neil b1854 (should be NY) but census says Ireland.
James O’Neil b1874 Brooklyn. Census gives age as 1yr 3mths.
Now I’ve cross-referenced other people from before and below on the census page to the city directories and addresses are 92, 94 and 98 Bergen Street so it’s reasonable to assume that 96 is in between? All the ages of the above are exact, the names fit and James’ occupation is correct. The only error is Hannah’s place of birth but I can live with that. Hanna’s full name was Hannah Angela O’Neil (nee Ames).
Then there’s William, age 3 born Brooklyn. I must confess that in nearly 20 years of research our family never knew anything about his existence. I can’t find any birth or baptism record but I’m not surprised to be honest as James and Hannah did not marry until April 1873 at St Teresa’s on Henry Street lower Manhattan so he must have been born out of wedlock. This is the only record I have of William so I’m assuming he died young.
James and Hannah were back in lower Manhattan with James' immediate family by October 1876 at 84 Madison Street. This had been and was the main O’Neil family home from 1863-1883 and he gives this address on his naturalisation with a prior address as 96 Bergen Street - so all the details here are absolutely confirmed.
I could go ahead and order both possibilities from the indexes one being very strong but I wanted to check the newspapers first to see if I could be certain before making any order:
O’Neil, Wm, Age 3, Mar 11 1876, Cert No. 231858, Manhattan.
I doubt the other family address of 113 E12th Street would feature but I include it just in case.