There is perhaps no city on Earth more mythic in emotional scale or more torn by conflict than Jerusalem. It is one of the oldest cities in the world, having been dated to as early as 4000 BCE. Next to Byblos in Lebanon, Jerusalem may be the oldest continually inhabited city in human history. It has been destroyed twice, once by the Babylonian Empire and then by the Roman Empire. It has been under seige, captured, recaptured and otherwise fought over hundreds of times. This makes the origin of its name tragically ironic.
At the root of Jerusalem's Hebrew name, Yerushalayim, are three letters. They are Shin, Lamed and Mem. In all semitic languages, that root pattern and its cousins have the same connotation. Shalom, Salaam, peace.
There are many early references to the city of Jerusalem. Biblically, the first mention of it is in the book of Genesis, chapter 33. As part of his travels, the patriarch Jacob journeys to a country called Shachem, specifically to a city called Shalem. This word, Shalem, essentially means "completeness" or "wholeness". As is often the case, specific locations in the Torah are used to represent philosophical concepts. If a troubled biblical figure travels to a city called "wholeness" there is some sort of lesson to be learned. From this and from other references, it's fairly obvious that the people of the ancient world saw some great importance in the city that would be Jerusalem. Exactly why, it is impossible to say.
It was not just early monotheistic societies who saw greatness in Jerusalem. Egyptian records, specifically the famed Amarna Letters, mention a city in the levant using the Akkadian term Uru Uru Salam Ki. That "salam" ought to be familiar by now. As for "Uru Uru", this shares an etymological root with a modern Hebrew word, Ir, meaning "city". This is where the "Jeru" in Jerusalem comes from. In ancient semitic languages, repetition was used as a means of emphasis or aggrandizement. The "ki" was just a grammatical structure declaring the end of the phrase. In this case, the phrase would mean "That great city of peace".
As Jerusalem grew and the language spoken in it solidified, the name transformed into its modern state. The current suffix "ayim" is an ancient Hebrew grammatical structure to denote duality. This makes reference to the fact that Jerusalem is built upon two hills. There is another frequent geographic term in Hebrew that uses the dual suffix. Mitzrayyim is the Hebrew word for Egypt. At the time when the Hebrew people interacted with Egypt, it was split into two kingdoms.
Many cultures have recognized Jerusalem as an important city and have used it as a symbol of peace and righteousness. The early American settlers of the Puritan pilgrim lines were especially fond of the concept of "the city on the hill". American cities like Salem, Massachusetts bear the same etymological root as the holy city. Only time will tell whether Jerusalem will remain such a pervasive symbol in human culture. Should our species ever colonize lands beyond our own planet, I wouldn't be surprised to hear of a settlement called Jerusalem.