Well...
It is a bit difficult to have this happen, as too many things were against it. Having Paul live longer is, in my view, almost impossible, since he had terminal prostate cancer by that time; Constantine II was rather young and inexperienced. This leaves the Queen Mother, Frederica, at the helm of the Palace /Royal faction. Frederica was not very prone to negotiations and had already clashed with the previous Prime Minister, Constantine Karamanlis, who was pretty much the leader of the main conservative party and rumoured to owe his assuming of the office of Prime Minister after the death of Papagos to a large extent to royal support; thus, with a more independent-minded Prime Minister, things would likely be worse.
Also, unlike almost all the surviving European monarchies, the position of the Greek royalty was uncertain: the monarchy had only been reinstated in 1935 after a rather rigged referendum, only for the royal family to be forced to leave the country 6 years later due to the German invasion and remain in exile for at least 3 years. And when they returned, their future was very unpredictable, as during the war, a parallel power player, the National Liberation Front, had consolidated its position in Greece, one of clearly leftist orientation and thus not very supportive of the monarchy and which enjoyed widespread popular support. It was only thanks to British support (initially) and later on American assistance that this threat for the monarchy had been largely eliminated, 5 years after the liberation of the country. All these made the royal family more apprehensive and also more interventionist : they wanted to govern/rule, not just reign.
This can be seen in the OTL conflict with Papandreou : the controversy over the ministry of Defence could be interpreted as the Palace's desire to control political life (usually, the Palace asked to nominate-unofficially of course - the Defence and Interior ministers, in order to control the army and the police better; sometimes, they asked the same for the Justice Ministry too, in order to be able to exert influence over the judicial system). Also, the "ASPIDA" (shield) affair was most probably an attempt to purge the army of officers potentially opposed to its potential use domestically by the monarchy or at least strengthen the latter's grip on it.
If Andreas stays in America as a professor of Economics, this would most certainly help his father, as it would remove the charge of George Papandreou attempting to cover up the ASPIDA affair in order to protect his son, which was the main argument used by Constantine to ask the former to leave the Defence Ministry. Also, if Andreas didn't return and subsequently rise quickly through the ranks of the Centre Union, the resentment of other founding members of the party (which was more of a big tent organisation) at the supposed show of nepotism, which could also threaten their prospects for the party's leadership and led many to side with the Apostasia governments wouldn't be there, at least for the time being. Thus, George Papandreou could secure the loyalty of his party more. But this doesn't remove the threat entirely: the anti-Papandreou groups could still bribe enough Centre Union members of parliament and perhaps engineer a change of leadership in the National Radical Union (NRU) in order to put a more cooperative leader in the place of Panagiotis Kanellopoulos (who wasn't willing to undermine the democratic institutions and was a more respectable and independent-minded figure than many would have liked) and thus facilitate the overthrow of George Papandreou.
Therefore, we need to avoid the conflict between Papandreou and the Palace. Perhaps have Papandreou accept the royal request for the Defence Ministry. This at least postpones the crisis. Then, perhaps have Makarios be somewhat more pro-West and avoid incidents like those in Kofinou in 1963; also have someone other than Dean Achenson take the lead in the talks about the Cypriot question and have that someone be more supportive of maintaining the status quo. If this is interpreted as being partial to Greece, then, combined with the ongoing discussions about moving closer to the EEC which proved to be fairly divisive, it could cause some instability in Turkey and perhaps strengthen the Islamists (and the nationalists) enough so that the army would have to intervene again. With Turkey busy on its domestic front, its government may not be able to push more regarding Cyprus and may accept a proposal providing for more protection for the Turkish minority in the island and the Americans providing funds for the development of the island. Thus, with a more stable situation in the island, a not - radicalised Turkish minority (thus no "pockets" policy in the 1960s) and Johnson and Papandreou not crossing swords over Cyprus, the Americans would have a better opinion of the Greek Prime Minister (Andreas could also help as a professor in the USA with a small PR campaign),while Papandreou would have scored a diplomatic victory.
All these would improve the Prime Minister's position back at home, which would make the more conservative factions more reluctant to turn on him. Furthermore, without the ASPIDA affair and the related problems, the King would have a much better relationship with Papandreou, which in turn would make him at least somewhat less receptive to the advice of his more reactionary members of his environment. If this relative tranquility wasn't completely momentary, then there is a chance of Papandreou remaining Prime Minister until mid 1966 or early 1967, as by then some problems would have most likely arisen.