Remeber that the research question is the center of a research project, it is fundamental as it focuses the study, determines the methodology and guides all the stages of the process. Here are some characteristics of a good research question, it has to be feasible, significant, clear and ethical.
In this case the political lens of history gives you a clue of the temes you can address, it needs relate the Irish immigrant experience to the government or public affairs of the United States.
A possible research question would then be:
How did the political ideas brought by the Irish immigrants affected the U.S. society in the early 20th century?