Thanks for your reply.
I don't mean to be rude, and I do realize you probably need to use a form response, but some of your suggestions would be very misleading for the unaware.
I clearly stated the era concerned - 1870s New York City. The suggestion to visit ellisisland.org is not valid. I know that, because I've been researching for a number of years. A newbie is going to waste time with a visit to ellisisland.org and go away frustrated, probably without ever realizing that Ellis Island did not exist in the 1870s.
I have a great deal more information, but I really did not think you would want a 5-page essay as an initial message. I've also found a great deal more information in the intervening weeks between my post and your reply.
I was specifically hoping to find a connection in Louth, because of the Brady-Donnelly obituary I quoted. From that death notice we know that Mary Brady was from Louth. We know that there was a Brady-Donnelly marriage, although we don't know where, Ireland or America. We know that Frank/Francis Donnelly is probably related to my William Donnelly - possibly either an older brother or an uncle - since they were living at the same address.
William Donnelly arrived NYC in 1872, with James and Margaret Donnelly - possibly siblings, but relationship was not stated on the manifest. Manifests before the opening of Ellis Island had far less information than the later manifests.
William Donnelly married Elizabeth Fetherston, also from Ireland, county unknown, in the USA, about 1873. He was naturalized in NYC in 1875 - hence the probability of 1852 as his year of birth, since he had to be at least 21 at naturalization. Arrival manifest and 1880 census point to 1852 as year of birth.
Elizabeth Fetherston Donnelly died in 1890 in NYC. In 1892, William married young widow Catherine McGarrity Nealis, from Inishative in County Tyrone, in 1892. They moved to Harrison, New Jersey, where William worked as a butcher for Swift & Co. They raised their blended family of Mary Nealis, Thomas Nealis, Anna Donnelly and added William Aloysius, Catherine, Patrick James, Sarah Lauretta, Margaret and Paul Francis between 1893 and 1905.
Most of the names of William's children are also probable siblings of William, found in the directory and naturalization lists of NYC, at the same or very close addresses.
William's mother, Catherine Molloy Donnelly, died in NYC in 1879. Her death certificate indicates that she arrived about 1866. As far as I have been able to determine, William's father died in Ireland before Catherine and assorted children left for America.