Do you know the names of any siblings who might have also immigrated? That might help you if there's a sibling with a less common name. I've got a bunch of Daniels and Patricks and Johns (oh the Johns!) that I'm tracking down but there's also a couple of Edmunds and Jeremiahs that help track the more common names. (Though I will admit, I am finding a lot more Jeremiahs than I would have expected!) So when I find John O'Leary and Jeremiah O'Leary together I look closer to see if they're the people I'm looking for.
I've also found knowing my ancestors occupation helps. Looking for teamsters and not bankers for example.