On this day in 1493, Christopher Columbus came across the island we now call Puerto Rico, dubbing it San Juan Bautista (named for St. John the Baptist). Before that, it had been called Borikén by the indigenous Taíno people, giving us the root of the word "Boricua," which Puerto Ricans regularly call themselves.
In 1521, Spanish colonists founded the Ciudad de Puerto Rico (meaning city of the rich port), which we now refer to as ... San Juan, the capital of the now-U.S. territory. So how did the name switch happen? It's not clear, but it definitely has something to do with the city's original and unwieldy name, the Ciudad de Puerto Rico de San Juan Bautista. Over time, that got shortened to San Juan, and the island somehow took on the first half of that name.
You can learn more about the history of Puerto Rico in this article from EBSCO.
