The Israelis have two options available for destroying Hizbollah. The first option is to get someone else to deal with it. That means either the Lebanese and/or the UN. Keep in mind exactly what Hizbollah is. It is a radical Islamic organization dedicated to the destruction of Israel, and the eventual establishment of a world-wide Islamic dictatorship (in cooperation with its patron, Iran). Hizbollah has taken control of about a third of Lebanon, and runs it as a religious dictatorship, a branch office of the Iranian religious dictatorship. Hizbollah's power base is the 1.3 million Lebanese who are Shia Moslem (like most Iranians are). The Shia comprise about 35 percent of the Lebanese population, and have long been the least prosperous third of the population. Hizbollah not only helped defend Shia interests during the 1975-90 civil war, but gave out tens of billions of dollars in Iranian money over the years. In return for all these favors, Hizbollah asks only for obedience, and volunteers for its trained terrorist force of several thousand fighters. Pro-Hizbollah Shia also dominate in the Lebanese army, a force put together since 1990 with the assistance of the Syrians. The Syrians are also allies of Iran, and consider most of Lebanon as part of Syria. France assembled Lebanon in the 1920s, after World War I, from bits of the recently disbanded Turkish empire. Historically, "Lebanon" was a string of coastal cities in what is now Lebanon. The French added some more territory inland, territory that had traditionally been considered part of Syria. The Syrians have not forgotten, neither have the Lebanese.
As part of the 1990 peace deal, brokered by Saudi Arabia, several divisions of Syrian troops were stationed in eastern Lebanon. These troops were necessary at first, but not for the last decade or so. The Syrians stayed to back up Hizbollah, make money by running the local economy, and because there was no one available to force them out. That changed last year, when years of anger at the Syrian occupation erupted into violent public demonstrations. The Syrians took the hint, and left. The 65 percent of the population that is not Shia (and is mostly Christian), are really unhappy about Syrian influence in Lebanon (the the murder of several Lebanese leaders over the last few years), and the continued existence of Hizbollah. But the Lebanese don't want another round of civil war, just to disarm Hizbollah. Since the Syrian army was sent packing, negotiations were under way with Hizbollah to disarm them, and return "Hizbollahland" to Lebanese control.
Hizbollah was split on the disarmament issue. Many Lebanese Shia wanted to become part of Lebanon, not a state-within-a-state. But the more hardcore Hizbollah believed in the goal of destroying Israel and establishing the worldwide Islamic dictatorship. The hardcore guys pulled off the July 12th kidnapping of two Israeli soldiers. That kind of operation was a violation of the unofficial deal the Israelis and Hizbollah had worked out over the past five years. That raid indicated that Hizbollah was no longer in control of all its fighters.
The radicals were now able to do whatever they wanted, including firing off the 12,000 or so rockets Iran had sent to Hizbollah over the last six years. These rockets were not intended, by Iran, for an actual attack on Israel, because such an attack would not destroy Israel, and could trigger an Israeli counterattack on Iran. While the Iranians publicly show contempt for any Israeli air strike on Iranian nuclear weapons research sites, their military people, and many of the politicians, know better. Iran was not happy when the Hizbollah rockets started flying, and Iran apparently made sure that the larger Iranian made rockets (that could reach most of Israel) were not used. That point has been missed by most observers of the war.
The Hizbollah attack left Israel with two options. They could either launch a massive invasion, and overrun all of Lebanon and Syria, or do what they did (to encourage the Lebanese and UN to deal with Hizbollah.) The trouble with the second ("small war") option is that it takes longer, and that leaves Hizbollah intact for longer. But the first ("big war") option would leave thousands of Israeli soldiers dead, and involve the occupation, for months, if not years, of Lebanon and Syria. That strategy would involve handing Lebanon back to its elected government with the understanding that there would be no more Hizbollah. But there would still be the a Shia minority, and within that minority there would still be Shia radicals who took orders, or at least direction, from Shia radicals in Iran.
Syria has to be overrun because, if you don't, Hizbollah can retreat to there from occupied Lebanon and set up shop in Syria. Take Syria and you eliminate any refuge (except Iran, where at least the senior Hizbollah people would flee to). While the Syrian military is no pushover, their armed forces have fallen apart since the end of the Cold War, and Soviet subsidies. Syria is a dictatorship run by the Alawite minority. The Alawites are, technically, a Shia sect, and for that reason, Iran subsidizes them. The majority of Syrians are Sunni Moslems. The Alawites have continued to run the nation because they established an efficient police state, and they get enough money from Iran to keep the ramshackle thing going. But the Israeli army could put the Alawites out of business in short order, and turn the place over to the UN for democratic elections (the first in nearly half a century). That would put Sunni Arabs back in power, and eliminate support for Shia Hizbollah.
There's one catch with Syria. Over the last two decades, Syria has invested some of its scan resources in one segment of its armed forces. As a result, Syria has a force of several hundred ballistic missiles, all of which can reach deep into Israel. Syria also has chemical weapons (nerve gas, and others). An attack on Syria puts Israel at risk of taking a few hits from Syrian ballistic missiles armed with chemical warheads. While Israel has its Arrow anti-ballistic missile system, a dozen missiles fired at once could overwhelm it. The risk is several thousand dead Israeli civilians, maybe more. But maybe none, if Israeli plans to take out the Syrian missile forces work. But in the aftermath of this Summers fighting, Israeli planners may have a new respect for possible deceptions and techniques for hiding missiles from attack.
The "big war" strategy has other costs. Mobilizing the entire Israeli armed forces means shutting down much of the Israeli economy, because so many key people are reservists. There is also the risk, however slight, of other Arab states declaring war on Israel. This risk is slight because those other Arab states are Sunni Moslem, and welcome the removal of Iran backed Shia entities (Hizbollah and Syria). But the risk is there.
There's always risk, it's a question of which one you estimate will do you the most good. Israel still has the "big war" option available, and Lebanon and Syria know it. If the small war option doesn't work out, Hizbollah, Lebanon, Syria and Iran know what comes next.
While Hizbollah, in typical Arab fashion, proclaims their defeat as a victory, Israelis are arguing over the merits of the two strategies available to them for eliminating Hizbollah completely. The recent fighting crippled Hizbollah military power, destroyed billions of dollars of its assets, and actually improved Israeli combat power. Thousands of Israeli troops gained combat experience in southern Lebanon, and Israeli casualties had no effect on overall Israeli military strength.