In the usual story, Venezuela is named after a city on another continent, because some houses on stilts reminded a sailor of it.
In 1499 an expedition led by Alonso de Ojeda reached the coast of what is now Venezuela, a year after Columbus had reached Paria. Amerigo Vespucci was aboard. Around Lake Maracaibo the Europeans saw villages of houses raised on stilts over the water.
Etymonline gives the standard version: the name is a Spanish diminutive of Venecia, "Venice," supposedly given in 1499 because of that village on stilts. Vespucci is usually the one credited with the comparison. In a letter later used by Martin Waldseemüller, he wrote: "We found a village built over a lake, like Venice."
The problem is the evidence behind that letter. Caracas Chronicles notes that the letter describing Vespucci's first voyage is now believed to be fabricated. The earliest record of the name itself is on a map drawn in 1500 by Juan de la Cosa, a companion of Ojeda.
A second explanation comes from Martín Fernández de Enciso. His Suma de Geografía, printed in Seville in 1519, was the first Spanish-language account of New World geography. He used "Veneciuela" as a local name: for the people themselves in one reading, for a village on the lake in another. Either way, the Spaniards may have borrowed a native name that only happened to sound like Venice.
A third claim, in a work by Antonio Vázquez de Espinosa, says Venezuela meant "big water" in the local language. The historian Tomás Straka says the native-origin theory "has never been backed by linguistic or ethnohistoric proofs."
The name first covered the Gulf of Venezuela. It became the Captaincy General of Venezuela in 1777 and survived independence.
The name of the model city has its own story. Etymonline traces Venice through Medieval Latin Venetia to the Veneti, an ancient people. The comparison stuck in German too: a failed German colony in Venezuela, run by the Welser bankers from 1528 to 1546, was known as Klein-Venedig, "Little Venice."