Robert father of William came over to the US about the same time William and his children did, but Robert's father Robert never did. According to Hardaman's History, see excerpt below. I am from Hugh Bodkin's Daughter Mary's line. Mary married Alexander McQuain. If you need more on this line please contact me.
The Bodkins of Galway, and the Earls of Desmond and Kildare, were descended from the common ancestor, Maurice Fitzgerald, Lord of Windsor, and one of the first invaders of Ireland, under Strongbow. His son, Thomas FitzMaurice, acquired ample possessions in Munster, where his descendants became Earls of Desmond. Richard, the son of Thomas, about the year 1242, held considerable properties in Connaught, under Richard de Burgo, and Thomas, his son, was the ancestor of the Bodkin family. This family name originated, according to tradition, from a victory gained by their great progenitor, Thomas Fitz Richard (about the year 1300,) over a valiant Irish knight, whom he encountered in single combat, and having, in the conflict, made use of a short spear or weapon, in Irish called, a Baudekin, he was, from that circumstance, surnamed, Buaidh Baudekin, of the victory of the Bodkin, which name was afterwards retained by his descendants. Whatever doubt may attend this traditionary relation, none can exist as to the origin and descent of the family, which are fully ascertained by the testimony of antiquaries, by ancient stone sculptures and monuments, still remaining, and from the genealogies of the Geraldines, whose arms the Bodkin family bore for many generations, and whose motto, Crom aboo, they retain to this day. [y] Henry Bodkin, the son of Thomas, was Clericus ville in the reign of Richard II. at which time, there was a street or lane in Galway, called Baudekyn's lane?. They were then possessed of large properties in and about the town, particularly at Newcastle?, near the river; [z] and at Athenry?, [a2] Toherskehine?, Ballynameathagh?, Kilcornan? and Parke?. At present the principal families of the name, are those of Annagh?, Carrowbeg?, [b2] Castletown?, Kilcloony? and Thomastown?.
There are more pages in the history that tells more of the story.