The manifest says her destination was New York. Other passengers gave destinations other than New York, so it didn't just refer to where the ship was arriving. Most of the destinations on her page were states, though, not cities (except for Chicago), so she might have been headed to somewhere in New York State. I'd bet on NYC though, at least to start.
https://www.familysearch.org/ark:/61903/1:1:QVPV-YZBFMaybe she stayed in New York. Does she show up on any city directories in NYC? Females were not listed nearly as often as men, but it might be worth a try for a few years after 1872.
I tried familysearch for anyone named Emma with a mother named Sarah E*dge, residence: United States, but didn't find anyone. You could try that at other sites and see what you find.
Did any known siblings or close relatives emigrate to the US? Could she be with them?
I tried familysearch for anyone named Emma, born in England between 1848 and 1851, living in New York, NY, and narrowed the search to leave out birth records. 731 Results. - You could go through them and eliminate all the ones that weren't wives, then all the ones who had children born in the US before your Emma was old enough to marry. Or before 1872 if you come to trust that passenger to being her. Then take a closer look at each.
I only looked at the first page of results, but by the time you get a few pages in, you might be able to eliminate all of them, as familysearch widens the field of possible matches - they might start showing people with a different given name and a middle initial of "E" for example. Or residences farther afield.
