There's a few related things going on here, I think:

• Almost certainly it had to do with one of the major motivations behind those who went on the First Crusade - the "liberation" of Jerusalem (and even more specifically the Holy Sepulcher). Moreover, even if his wasn't the case for the majority of the participants at the very outset in 1095, Jerusalem became central to the enterprise as those who participated "read backwards" to try to explain what had happened. For instance, look at the titles to all the contemporary accounts of the First Crusade -- History of the Journey to Jerusalem, Deeds of the Franks and Others on their Journey to Jerusalem, History of the Expedition to Jerusalem, etc.
• Although "kingdom of Jerusalem" evoked/ alluded to the biblical kingdom of Israel, you couldn't (as a 12th-c Christian) recreate it because it had been destroyed by God via the Romans, and also could only legitimately be recreated by either the antichrist (bad) or Jesus Himself, just after the antichrist was defeated. "The Kingdom of Jerusalem" therefore links it to the past but signals that what they're creating is slightly different from what came before -- different and better, because Christian.
• Finally, the Latin for "kingdom" is not really so straightforward. Usually, we translate regnum as "kingdom" but it's really not that straightforward. It can often (and better) mean something more akin to "territory over which someone has power." This means that what really seems to be going on is that there's a "king" (rex) of Jerusalem who possesses a regnum. Later translators, particularly in the 19th and early 20th centuries, attached that regnum to something related to the land. They did this all the time. Same thing for "France." Up through the 12th century you had a "king of the Franks" (rex Francorum) who ruled Francia. Sometimes you had a regnum Francorum, but that referred to the territory of the Franks -- people, not territory. So, in other words, the territory itself didn't have a name (well, it did, but in the sources it's really a hodge-podge of biblical and classical references).

It's worth exploring how the position of King of Jerusalem came to be to explain some of why this happened. It was really not clear at the end of the First Crusade how the newly conquered territories would be governed. In theory, none of them were answerable to any of the others. There was no reason for Antioch to answer to Edessa or for either to answer to Jerusalem.

Godfrey of Bouillan was chosen to be the first ruler of Jerusalem. He refused to take the title of King, preferring instead something closer to 'protector'. He was chosen in no small part because he was the first major Crusader to go over the walls of Jerusalem during the siege (his siege tower had reached the walls with him on top of it). Godfrey was immediately set upon with a major problem, though, as a group of Crusaders reached the city soon after his election to complete their pilgrimage. This group included Bohemond and Baldwin, the rules of Antioch and Edessa respectively, as well as Daimbert of Pisa, a papal Legate. In theory these three very powerful lords could have upset or challenged Godfrey's rule. Godfrey appeased Daimbert by ejecting another bishop, Arnulf, from the position of Patriarch of Jerusalem and giving Daimbert the position instead. Bohemond and Baldwin were generally distracted and appeased and returned to their territories without causing any trouble.

Then, in 1100, Godfrey died and there was a bit of a race to see who would replace him. Godfrey doesn't seem to have explicitly declared an heir but it was generally known that he favoured his younger brother, Baldwin of Edessa, for the position. In contrast, Daimbert wanted Jerusalem to be controlled by the Church and to construct a new ecclesiastical state around the city. Unfortunately for Daimbert he was off besieging Haifa with Tancred (another Crusader) when Godfrey died and while he was away the deposed but still around bishop Arnulf, along with other supporters of Godfrey, seized the major fortifications of Jerusalem and sent word to Baldwin to rush to the city. Baldwin did indeed get to Jerusalem as fast as he could, fighting a small Muslim force along the way, and made it to the city before Daimbert could stop him. Baldwin was elected ruler of the city but not made King, since a king requires a coronation. The coronation only came on Christmas Day in 1100 when Daimbert agreed to crowning Baldwin in exchange for keeping his position as patriarch. Baldwin thus became Baldwin I, king of Jerusalem.

When Baldwin first became king the area controlled by Jerusalem was quite small. Crusader power extended to a few of the villages around Jerusalem (Bethlehem being the most famous) but control of the region was very limited and Muslim raiders a frequent problem. Over the course of the period 1100-1110 Baldwin I slowly expanded the area controlled by Jerusalem by taking new cities and extending the influence of his regal authority. The actual process involved in this is really complicated and so I'm pretty grossly oversimplifying things. Baldwin didn't necessarily take every city personally (Tancred was very influential in this process, for example) and it wasn't all successes either. By and large, though, Baldwin I and his successor Baldwin II expanded royal authority from their seat of power in Jerusalem over most of the Crusader states. Baldwin managed to keep Edessa generally under the control of Jerusalem since it was originally his county and still in the hands of his family. Antioch traditionally bucked what Jerusalem wanted it to do but the two territories did sometimes ally for major campaigns. Eventually Jerusalem was the more powerful of the two and so it was harder for Antioch to resist doing what Jerusalem wanted (that doesn't mean they couldn't though!)

I guess the short version is that the position of King initially only really applied to the city of Jerusalem, hence King of Jerusalem, but over the years those kings expanded their influence into the surrounding territories thus constructing a Kingdom of Jerusalem.

Thomas Asbridge has a good chapter on the creation of the Crusader States in his book The Crusades while Malcolm Barber's The Crusader States is the better work for a really detailed history of how the various political alliances that made up the kingdom of Jerusalem came to be.

when the city itself had no precedent for denoting the larger area?

Well, you're incorrect here. While it could be argued that the city had no historical precedent as such, it does have scriptural precedent.

http://biblehub.com/revelation/21.htm

I saw no temple in [the New Jerusalem], for the Lord God the Almighty and the Lamb are its temple. And the city has no need of the sun or of the moon to shine on it, for the glory of God has illumined it, and its lamp is the Lamb. The nations will walk by its light, and the kings of the earth will bring their glory into it. In the daytime (for there will be no night there) its gates will never be closed; and they will bring the glory and the honor of the nations into it; and nothing unclean, and no one who practices abomination and lying, shall ever come into it, but only those whose names are written in the Lamb’s book of life.