The most common guess about the Golden Gate is wrong. The strait is not named for the California gold rush, and the bridge is not named for its color.
The name came first. Until the 1840s, the narrow entrance to San Francisco Bay was known in Spanish as the Boca del Puerto de San Francisco, "the mouth of the port of San Francisco." On July 1, 1846, the Army officer and explorer John C. Frémont looked at the strait and gave it a new name. He later reported saying that it was "a golden gate to trade with the Orient."
Frémont did not use plain English. In a geographical memoir submitted to the U.S. Senate in 1848, he wrote that he called it Chrysopylae, Greek for "Golden Gate." He explained the choice by comparison: the harbor of ancient Byzantium had been called Chrysoceras, or Golden Horn, and he gave his strait a name for the same reasons.
The Greek is easy to take apart. The prefix chryso- comes from khrysos, "gold," and Etymonline notes that scholars usually treat that word as a loan from a Semitic source. The second half is the Greek word for a gate. Frémont was thinking of a doorway through which wealth would arrive from Asia, in the way the Golden Horn had once brought ships and riches to Byzantium.
The "horn" in Golden Horn refers to the inlet's shape. Wikipedia says the meaning of "golden" is more obscure. One idea is the riches that flowed into the city through its harbor. Another is the yellow light on the water at sunset. Frémont's own gate was about trade, which fits the first idea.
The name then got an extra layer of meaning. Gold was found in California in early 1848, about a year and a half after Frémont's christening. The Golden Gate Bridge organization says Charles Preuss's 1848 map was the first to show "Golden Gate" at the bay's entrance, and by then the name looked like a prophecy.
The Golden Gate Bridge borrowed the strait's name. Construction began in 1933 and the bridge was finished in 1937. It did not create the name. It only made it famous.