I've done a lot of research on the US Coits. Pretty much they all descend from one man, John Coite b abt 1592 who arrived between 1630 and 1635 in Salem. He supposedly was Welsh and evidently sailed from Chepstow. I've been to the town of Coitie in Glamorganshire but didn't find records, although my research time was limited. A Welsh researcher checked the tax roles in Wales and couldn't find any Coits there today. John Coite states in his will that three children were left behind and he knew not if they were still living. http://www.wikitree.com/wiki/Coit-6
The family has been associated with the "Welsh group" who moved to New London CT with Rev. Blynman in 1650 and were granted lots in the same area. But at least one of that group, Hugh Calkins, who was always assumed to be Welsh, has recently been proven to be English. The confusion evidently comes from those early records showing he sailed from Chepstow, a Welsh port. See Calkins Family research link on Hugh Calkins profile.
There was a Puritan Thomas Coit b England in 1600 found in Leyden in the 1620s who married Priscilla, the daughter of Moses Fletcher (Mayflower). There is a reference to his brother John as a witness to his first marriage in 1623 in Leiden. Could this be John Coit b. 1592? Thomas named his first son John. And in the US Coits, there were several Thomases although not in that first generation. Could this be where John Coit left his oldest three children? There was an English Separatist community there in the early 1600s, including Brewster. Anyone up for some research? Here's a link to the book where John Coit is referenced in a footnote: http://books.google.com/books?id-EwiqC6neusC&qcoit#vsnippet&qcoit&f=false
There is a branch of that Dutch Coit family today, descendants of Thomas and Priscilla, who use the spelling Koet and Van Koet. A Y-DNA test with one of our US Coits and one of them should tell us if indeed John and Thomas were brothers. We know they were both Puritans. One born 1592 and the other 1600 could reasonably be brothers.
The above stated, I have heard that there was a Portuguese Coity (sp?), who arrived in the US in the 1800s and adopted the Coit spelling. I have never come across them in my research, but they may exist.
There are some English Coites, using the Coite spelling. Families researching them can't seem to get beyond the early 1800s in the records so we can't connect them to John Coit 1592. They seem to be seamen. The US Coits were boat builders and seamen.
I have never heard of any connection to Ireland. I would love to know if anyone can further this research and learn the origin of the family.