The name Jayme originates from the Iberian Peninsula, most likely from the medieval Spanish and Portuguese form of the name James, which was derived from the late Latin name Iacomus, itself a vernacular form of the biblical name Jacob. The name Jacob comes from the Hebrew name Ya'aqov, meaning "supplanter" or "one who follows".

The biblical figure Jacob was the son of Isaac and Rebecca, and the father of the twelve tribes of Israel. The name Jacob is mentioned numerous times in the Hebrew Bible, also known as the Old Testament. The New Testament also refers to Jacob, as he was an important figure in the lineage of Jesus Christ.

One of the earliest recorded uses of the name Jayme can be found in the historical records of the Kingdom of Aragon, a medieval kingdom located in modern-day Spain. In the 13th century, Jayme I, also known as James the Conqueror (1208-1276), was the King of Aragon, Valencia, and Majorca, and is considered one of the most important rulers in the history of the Crown of Aragon.

Another notable figure with the name Jayme was Jayme Huguet (c. 1412-1492), a Spanish painter from Valencia who was active during the Renaissance period. His works can be found in various churches and museums across Spain.

In the literary world, Jayme de Burgos was a 15th-century Spanish poet who wrote in the tradition of the cancioneros, or poetic anthologies popular during the Middle Ages.

Moving to the Americas, Jayme Caravallo (1619-1695) was a Portuguese-born explorer and one of the first European settlers in the area that is now known as the state of Paraná in Brazil.

Later in history, Jayme Sidles (1857-1948) was an American educator and the first president of the University of Idaho, serving from 1892 to 1917.