"MY second cousin in Ireland is a widow. She is 42 and has a nine year old daughter. Her husband died three years ago.For the past six months she has been corresponding with an American man who has visited her a few times. He seems very nice, and they are talking about starting a life in the U.S. They are not engaged, but my cousin is planning on spending some time here in the summer to investigate the lifestyle and schools. Do they need a special visa for this? Also, I'm sure as an American my cousin's boyfriend would be able to sponsor her for a green card, but what about the daughter?"
YOUR cousin and her daughter will not require a particular visa to spend some time here in the summer. They are coming to investigate possibilities and see what life is like here, and as such they can enter the U.S. for up to 90 days as part of the visa waiver program which Ireland is a part of. In order to avail of this, they will need biometric Irish passports.
As far as legal status for your cousin's child should she marry her American partner, he will be able to sponsor the daughter for a green card. Step-parents can confer the same immigration benefits on step-children as natural ones, provided that the marriage that created the "step" relationship took place before the child turned 18 years old.
The applications for your cousin and her daughter can be filed at the same time after the marriage, and both would be considered immediate relatives under immigration law, which means that that waiting time for visa processing is significantly less than those being processed in accordance with other family-based categories.