Carol, “Patrick “ seems to be a pretty popular name among the Mones. In a New York City directory (1878) there is an Annie E. Mone, widow of Patrick, living on 279 Monroe.
In another New York record, a probate record in 1871, a Patrick Mone has the following descendants:
His widow:: Rosanna Mone; Son: James Mone, declaring hat he is the executor of Patrick’s will; Son: Frank Mone; Son: John Mone; All residing in the City of Brooklyn
Daughter: Ellen, wife of John Brophy, living in the City of New York; Son: Patrick, residing in San Francisco
A New York passenger list has a Patrick Mone arriving in the city on 1 May 1863, place of origin: Ireland; port of departure: Liverpool; ship name: City of Manchester
Hope these help some.