This is very hard, but not impossible.
1) The basic elements of democratic government, as the West understands it, does not exist in Middle Eastern culture, although some elements could be jury rigged to fit. So Westernization needs to happen to some degree.
2) To achieve "modernization", a strong central state was needed which meant reformers always strengthened autocratic rule first and overrode anything which could have served as the basis of a native democratic state. This is what derailed everything IOTL.
3) The Middle East lacks a large enough middle class for sustained democratic politics. So this needs to grow and become substantial.
4) The borders imposed after World War I lacked legitimacy which complicated state building and a cohesive internal culture. However, this only applies to Iraq and Syria, really and not to other countries.
5) The creation of Israel and the Arab's continued defeats by it kept de-legitimizing the Arab states who kept responding with internal repression to stay in power.
6) Superpower rivalry in the Cold War always meant any group of would be dictators could obtain support from one or the other superpower.
I don't like going all the way back to Sykes-Picot. Not only is it not realistic that you can keep the Europeans out, I don't think there is a definite and obvious line of thought that can establish a stable and democratic Middle East. It just allows fiction writing.
The key is Egypt. It is one of the leaders of the Arab world, and was the paramount leader for several decades. If Egypt can become democratic, it can act as the mortar for the rest of the region.
Egypt did have a "liberal" period of parliamentary democracy under the monarchy. However, it suffered from the ineffectual monarchy of King Farouk, the continued presence of the British, and the government being discredited in war with the victory of Israel in 1948. This is what allowed the Free Officers to overthrow him in 1952 and eventually establishing a republic in 1953 which lead to an authoritarian dictatorship (albeit an extremely popular one for several decades). Nevertheless, during the time of the monarchy, there were elections, some form of parliamentary rule, and a building of the state.
I think if this could have been continued in Egypt, that Egypt could eventually become a democratic force in the region and with its size, population, and cultural center, it could have supported democratic transitions elsewhere, especially within the more conservative monarchies.
The first step is to defeat the Free Officers coup. Let's say Farouk learns of the coup and arrests the officers beforehand. Just to make things easier for us, let's say Nasser is killed. He is the most dangerous to Egyptian democracy since he was very charismatic, but a committed enemy to it.
This is when it gets harder. What is needed is for the constitutional monarchy to start reforming in important ways to renew support for the government - passing land reform, defying the British, economic development, and bringing the new urban middle class into the democratic system. So it is some combination of what allowed Nasser to become popular, but retaining parliamentary democracy. It is very debatable if this can actually happen, but let's assume it does.
Without the example of the Free Officers Coup, then similar incidents that deposed the monarchy in Iraq don't happen.
US support for Egypt grows as British influence diminishes. Economic aid flows into the country to build anti-communism. Over a period of several decades, democratic institutions strengthen although Egypt isn't consistently pro-West. However, Egypt's example of constitutional monarchy and parliamentary rule prompts reform in Jordan, Libya, Iraq, and parts of the Gulf. Egyptian leadership is challenged to a degree from Wahhabist Saudi Arabia and a more radicalized Syria, but not substantially.
Despite ongoing problems with Israel, there is no second Arab-Israeli war. Gaza remains part of Egypt, and the West Bank is ruled by Jordan. Eventually through diplomacy in the 1980s, the conflict is partially resolved by Arab acceptance of Israel, allowing Jews to visit the Old City of Jerusalem, and compensation for the Palestinian Arab diaspora.
Arab democratic institutions are weak for several decades, but by the 1980s prolonged experience, institutional reform, and greater competence has greatly strengthened the institutions of democracy throughout the Arab monarchies. Without "Arab Socialism" their economies are stronger, less reliant on subsidies, and offer more opportunity than the Arab economies IOTL.
There are ongoing problems especially with Islamic extremism and violence as a result of the Muslim Brotherhood, Saudi export of Wahhabism, and perhaps the Iranian revolutionaries if that still happens. However, it is containable since the Middle East overall is less hostile and more prosperous. The appeal of Islamic radicalism is much less. In the past several decades, there has been no Qaddafi, Assad, or Hussein.