Well, I think that there's a tendency to overrate the American-Israel friendship a bit. I suppose this is because a) politicians often talk of the relationship and b) Israel gets a lot of criticism. I'm not sure if I'd define the American-Israeli relationship as closer than the American-Japanese or American-South Korean alliance. In fact, Israel is one of the few American allies that the United States doesn't defend or have a collective defense treaty with.

The American-Israeli relationship did not reach the current level of friendship until after the six-day war. Britain & France (mostly France) were originally the backers of Israel. De Gaulle kind of moved away from this in the 1960s. When the Europeans moved away, the United States swooped in. As I'm sure you know, the United States is very involved in the Middle East and has maintained strong alliances with Saudi Arabia & Turkey as well (Egypt too eventually). Having a strong ally in Israel that exists as a stable country is an effective thing. Israeli intelligence operations have been very useful. Israel managed to get their hands on a Soviet MiG-21 and turn it over to the United States. A lot of intelligence figures have said that Israeli contributions in intelligence are worth more than the aid.

Also of note, Israel is a liberal democracy with a high standard of living. Any Westerner can go to Tel Aviv and feel familiar. 90% of Israeli Jews like the United States. It stands on far more solid ground than the U.S. alliance with Jordan or Egypt or Saudi Arabia or Turkey, where most of the population dislikes the United States. Also, there's a large diaspora population in the United States and Christian evangelicals love Israel. I think that has more to do with why Israel gets talked about by politicians, but I don't think there's any evidence for it driving U.S. policy.

I do not feel their is any tendency to overrate the unique relationship that the U.S. and Israel have. To give an adequate answer I still start quite early. There were large waves of Jewish immigration from Europe in the late 19th century and early 20th century as "antisemitism" was prevalent in both Western and Eastern Europe. Many of these Jews came from Eastern Europe where pogroms (anti-Jewish riots) were common and violent. These Jewish immigrants were urbanized and were able to assimilate quite well. Simultaneously, there was Jewish immigration to Palestine under the Zionism movement or Jewish Nationalism. (Also a result of the antisemitism) This immigration was done under the British mandate. (British and French being the two dominant powers in the Middle East at this time; essentially carving up territories that were under Ottoman rule pre-WWI.) Later restrictions on immigration were put in place and would not be lifted for several decades. Post-WWII saw the diminishing strengths of the colonial powers of old and the rise of the U.S. and Soviet Union as the worlds chief superpowers. U.S. foreign policy focused on containing the spread of communism and limiting Soviet spheres of influence (Truman doctrine). This was especially important in areas key to U.S. national interests (Middle East). A debt stricken Great Britain, who was losing much of her colonies to nationalist movements, was all too happy to dump the Palestinian-Israeli conflict onto the U.S. UN resolution 181 (and the U.S. being the first to recognize Israel in the global scene) and the subsequent, distinct Arab-Israeli wars ended with the establishment of the state of Israel. For the war effort, future PM of Israel Golda Meir raised around $50,000,000 for weapons from Jews within the U.S., which demonstrates the power of the diaspora through private donation.

Israel has been the largest recipient of U.S. aid and for several decades has been receiving over $2 billion a year not to mention private donations. Much of the aid is military aid, in effect to always keep Israel economically and military stronger than all of its neighbors. These measures are seen in the 1967 and 1973 wars (including a strategic airlift in the latter). Interest groups, such as AIPAC have also had substantial influence and pressure in the relationship. Even with the fall of the Soviet Union, the U.S. uses Israel as a stabilizing hegemony in the region and despite numerous disagreements there is much cohesion between the two. Sorry if this seemed incoherent I tried to answer your question as best as possible.

Source: Gelvin, James L. The Israel-palestine Conflict : One Hundred Years of War. 2nd ed. Cambridge [England]: Cambridge University Press, 2007.