People will rightly likely answer this with a focus on the Iranian coup. One good thread on that is here. But from a narrative standpoint, the Suez Crisis was very important in the Eisenhower administration for numerous reasons (note that it was obviously more important for the French, British, Israeli, and definitely Egyptian governments, but we're focused on Eisenhower here). Happily, I also touched a bit on the Suez Crisis here. But, I'm guessing you're asking the question here so you can have some answers here rather than clicking through some links and other threads.
Part of the answer to your question is tied to whether believe you Eisenhower was being sincere or managing his image. David W. Lesch, for example, offered a scathing condemnation that the Eisenhower administration had such a poor understanding of domestic dynamics in Middle Eastern nations like Syria, that its interventions and actions in the region were failures layered upon failures. Keep in mind that under Eisenhower and Dulles, the State Department floated ideas and proposals all the way to the desks of Eisenhower and Dulles which in large part lay bare the fundamentally short-term mentality of Eisenhower's perspective on relations with the Middle East in general.
For example, one of the proposals Eisenhower and Dulles considered just to make Nasser feel isolated included manipulating the price of cotton on the international market in order to damage Egypt’s economy, building more facilities across the Middle East to jam Egyptian broadcasts, and working with the British to overthrow the government in Syria and installing “one more friendly to Iraq and the West.” Yes, you read that correctly. The government in Syria was, for a time, considered for overthrow not because it was destabilising the region or a threat to U.S. interests, but because it refused to comply with concerns the British had about Syria-Iraq relations. (for the quoted bit see Memorandum for the President, March 28, 1956, 1-3, Mar ’56 Diary, Box 13, Diary Series, Ann Whitman File, Dwight D. Eisenhower Library.)
Eisenhower and Dulles also really quite liked King Saud (Saud bin Abdulaziz Al Saud), because he was friendlier to the West, but also because he was very openly devoutly religious. Indeed, King Saud's foreign policy was in some part predicated upon an attempt to unify all Muslims. Put aside the viability or sincerity of that, having a foreign policy in part based on a religious drive meant to Eisenhower and Dulles that Saud was definitely not vulnerable to godless communism.
How much did Eisenhower like Saud? Eisenhower suggested that if the Middle East "had" to have a unifying Arab leader, instead of Nasser, the U.S. could throw support behind Saud and Saudi Arabia as the regional leader - for that see Memorandum of Conference with the President, March 28, 1956, 2, Mar ’56 Diary, Box 13, Diary Series, AWF,DDEL).
And of course, we should at least touch on the Eisenhower Doctrine. Note that by the time Eisenhower discussed the Doctrine with Congress, he was also arguing that the UN could not stop Soviet aggression, as shown by the Hungarian Uprising, whereas France and Britain had shown their “decent respect for the opinions of mankind” in adhering to the ceasefire in Egypt post-Suez. (Dwight Eisenhower, Special Message to the Congress on the Situation in the Middle East, January 5, 1957, The Public Papers of the Presidents of the United States: Dwight D. Eisenhower, containing the public messages, speeches, and statements of the president, January 1 to December 31, 1957, (Ann Arbor: University of Michigan Library, 2005).
But, we should keep in mind that Eisenhower never envisioned the Doctrine as actually requiring U.S. action or any kind of binding agreement or statement. Indeed, by February of 1957, Eisenhower was writing to Dulles requesting the means to respond to Middle Eastern leaders hoping to take advantage of the Doctrine in such a way as to avoid saying anything “specific or say anything that might tie our hands later.” (Memo, Eisenhower to Dulles, February 6, 1957, 1, Lebanon (3) Box 7, International Series, AWF, DDEL).
This is a very broad way of me saying that Eisenhower did pay close attention to the Middle East (in my opinion, when the region was causing "problems"), but that I think even in that speech you get an idea of the context and just how much attention.
References
Dwight Eisenhower, Special Message to the Congress on the Situation in the Middle East, January 5, 1957, The Public Papers of the Presidents of the United States: Dwight D. Eisenhower, containing the public messages, speeches, and statements of the president, January 1 to December 31, 1957, (Ann Arbor: University of Michigan Library, 2005
Memo, Eisenhower to Dulles, February 6, 1957, 1, Lebanon (3) Box 7, International Series, AWF, DDEL
Memorandum for the President, March 28, 1956, 1-3, Mar ’56 Diary, Box 13, Diary Series, Ann Whitman File, Dwight D. Eisenhower Library.
Memorandum of Conference with the President, March 28, 1956, 2, Mar ’56 Diary, Box 13, Diary Series, AWF, DDEL.
Richard J. Alexander. "Couscous Mussolini: US perceptions of Gamal Abdel Nasser, the 1958 intervention in Lebanon and the origins of the US-Israeli special relationship." Cold War History Volume 11, Number 3 (August 2011): 363-385.
David W. Lesch. Syria and the United States: Eisenhower’s Cold War in the Middle East (Boulder: Westview Press, 1992)