The McCaughey one-name study was registered in 2021.
It grew out of my interest in my own branch of McCaugheys, who originated in County Tyrone, Ireland and immigrated to Rhode Island in 1873. I then began becoming interested in other McCaugheys in the United States.
In every US census from 1870-1940, Rhode Island has had a higher percentage of McCaugheys than any other state.

McHaughey, MacCaughey, McGaughey

Despite its similarity in sound to (Mc)Coffey or Coffee, the "roots" of this name are quite different, the (Mac)Caughey clan being from the Ulster region. The Gaelic spelling is MacEachaidy, "Eachaidh" being usually Anglicized to Aghy or Oghy, and as such it remains a rare Christian name in modern Ireland. The precise translation is uncertain, but it is probably "the son (Mac) of the youthful one", "Eachaidh" being equivalent to the lad, or boy, or similar. 

The McCaugheys are descendants of the ancient Dál Fiatach dynasty, rulers of the Ulaid. 
The Dál Fiatach are claimed as being descended from Fiatach Finn mac Dáire, a legendary King of Ulaid and High King of Ireland, and are thought to be related to both the Voluntii and Darini of Ptolemy's Geographia. They are also perhaps more directly related to the pre-historic Dáirine, and the later Corcu Loígde of Munster. 

It is said that as MacCahee, the clan was recorded in County Tyrone in 1685, however, in the modern forms early recordings include: John McGaihy (1719); Elizabeth McGahy (1711); Jane McGahey (1846); John Caghy (1791); Jennet McCaughey (1816); Lettitia McGaghey (1829); and Margaret McCaughey, of Aughnacloy, County Tyrone, on August 5th 1868.

The most notable current members of the McCaughey clan are the McCaughey (pronounce "McCoy") septuplets, born in Iowa in 1997. 

According to forebear.io, "McCaughey"  is the 133,798th most common surname in the world, as of 2014.

Again according to forebear.io, as of 2014, the name is most prevalent in the United States, where 79% of the world's McCaugheys are found, but has the highest density in Northern Ireland, with a frequency of 1:1,894.

I've collected occurrences of the names McCaughey, McCaughy, McCahey, and McCahy, in:

US Federal censuses from 1790-1840

Canadian censuses from 1841-1891

Immigration records for the port of New York from 1820-1891

Immigration records from Ellis Island from 1892-1924

Immigration records from the port of Boston from 1848-1900

Immigration records from the port of Pennsylvania from 1800-1906

The censuses of England and Wales from 1841-1901

The 1901 census of Ireland (950 names)

The 1911 census of Ireland (860 names)

Griffith's Valuation of Ireland from 1848-1864

Irish Tithe Applotment Books from 1823-1837

The Irish Flax Growers List from 1796