Thomas Halligan emigrated to America from Ireland around 1848. There are a couple of records that match the time frame, but since we have no concrete information about his birth or his parents, it's kind of a toss-up as to which could be him. The dates on the passenger lists would put Thomas born around 1820.
The first is from the New York Passenger and Immigration Lists for the Ship Rappahanock. It's point of departure was Liverpool, and his place of origin in listed as Ireland. It arrived in New York on 1 May 1848.

Thomas Halligan, 4th name from the bottom, is listed as age 28 and a Laborer.
The second is from the New York Port Arrival Records. He sailed on the ship Izaac Wright. It sailed from Liverpool and arrived in New York on 26 Feb 1849.

Thomas Halligan, number 329, was listed as age 27 and a Laborer. Is is also possible that the Richard Halligan, number 326, age 25, may be Thomas' brother.
The 1850 Censes for Flatbush, Kings County, New York lists a Richard and Thomas Halligan (lines 40 and 41), ages 29 and 24, both Laborers, both from Ireland. Could be the same?

According to my Gpa, Don Caman, "The Halligan family and the Sammons had come from Ireland, they were in the famine and it was to either leave Ireland or starve to death. So, that's why they left Ireland. On her [his mother's] side they were all Irish. The Halligans and Sammons eventually settled in Iowa and my great-great-grandmother met my great-great grandfather in Iowa. My grandparents met in Iowa, they were born in Iowa."
Thomas Halligan was the father of Anna Teressa Halligan, who was the mother of Annie Marie Sammon, who was Donald Caman's mother. On the 1930 Census Anna Halligan Warren (2nd marriage) lists that both her parents were born in Ireland.

Annie Marie Sammon Caman

Annie and Don Caman, circa 1920
There is also the 1870 Census that lists Ann Halligan, 13, which matches her birth date of 25 December 1857. On this Census, her parents are listed as Thomas Halligan, 28 and Lancia Halligan, 40. The dates are weird, but it is the closest I have been able to find to link an Anna Halligan to a Thomas Halligan. It would put Thomas Halligan's birthdate in 1842, which is way off the time line for the passenger lists.

So, still a frustrating mystery.
Direct line ancestry: Thomas Halligan --> Anna Teressa Halligan --> Annie Marie Sammon --> Donald William Caman --> Donald William (Caman) Marshall --> Lora Teressa Marshall Nehring