Cui
A rare surname in the US, likely brought in from a European place-name or family-name tradition, though its exact origin is not secure from the available evidence.
The #9,251 most common surname in the United States (2010 Census) — about 1 in every 87,712 Americans shares it.
Etymology
The excerpts do not give a clear standalone explanation for Cui itself. They do, however, point to medieval Norman forms such as de Cui and a connection with a Norman place or fief named Cuy/Cohy, which suggests a locational origin rather than a descriptive word. In historical surname work, such forms often mean "from Cuy" or "of Cuy," but the evidence here is indirect and the exact form may be affected by OCR noise.
History
The historical material ties the name to medieval Normandy and to early recorded bearers such as Robert de Cui in the 12th century and later English occurrences of a similar form. The surname is very uncommon in the modern United States, so present-day bearers may reflect later immigration, variant spellings, or separate origins in other languages. Because the excerpts are incidental rather than a full dictionary headword, the etymology should be treated cautiously.
Origin
Regions
Variant spellings
Fun fact
One of the excerpts places a bearer of the name in the 12th century, showing that this tiny surname has a very old documentary footprint.
From the historical record
“Hagode Coarteney, co. Devon. 1273. A. iohn de Cui tenay, co. Soms., ibid, lenry de Curteneye, co. Soms., ibid. Hug^ de Coarteneye, ca Bedf., 20 Cdw. Egrlina de Curtenay, co. Ozf., Hen. III-Edw.I. K,”
— A Dictionary of English and Welsh Surnames (alternate scan)