Here's one idea...
In 1987 in OTL the Turkish vessel named Simsik started looking for oil in the Aegean. Andreas Papandreou stated back then that if the vessel is found within territorial waters it is to be sunk.
So, what if the Turks do not ignore Papandreou's warning and try to go on with their search for oil. Plans had already been worked out by the Greek Air Force on how to sink the ship and hit its accompanying destroyers.
So it's 1987, the Cold War is still on and the Greeks have just sunk Sismik together with 2 turkish destroyers.
You have retaliation from the turkish side, involving bombings of forward airfields in greek islands, Greece fights back, Greek+TurkishF4s and Greek Mirage F1s clash in the Aegean skies. In the Evros front, the Turks try to break through, stopped cold by Greek anti tank power and bad terrain. The Turks score major hits when they try and actually manage to capture several smaller islands in the Eastern Aegean. Their biggest gamble however is Rhodes and Kos. A big armada of landing craft exits Izmir harbor, while the bulk of the turkish fleet comes out of the Dardanelles. Greek subs (Type 209) score major hits on them, but the Greek navy is repeatedly hit by the Turkish Air Force. The Turks manage to get some forces to Rhodes and Kos, although the majority of their troops drown when their ships are sunked during the journey southwards. The Greeks manage to get their act together and after reinforcements arrive per commercial ferries from Athens, they establish good defence on both islands, too strong for the Turks to break.
It's a victory for the Greeks in Rhodes, but the recapturing of the smaller islands proves to be quite difficult, with the Turks bringing in reinforcements in by helicopters. Both air forces are quickly absorbed in a war of attrition and the Turks dont manage to use their newly acquired F16s.
The Cyprus Front is largely quiet during the whole war with the Turks probably hoping to take care of Cyprus once they strike a decisive victory against Greece.
The war goes on for a week, before the USA enforce a cease-fire and return to the status quo before the war. This is accomplished by the USS J.F.Kennedy and USS Saratoga battle groups joining up and making an joint entry into the Aegean Sea 8 days after the conflict started.
On a side note...
The "Papoulias visit": In OTL at the climax of the Sismik crisis, Papandreou sent his foreign minister (Papoulias) in the middle of the night by plane to Sofia, Bulgaria. His visit was made public to the rest of the world and the Americans warned the Turks that the Greeks may have tried to persuade the Bulgarians (Soviet puppet state at that time) to open a front against Turkey, should war commence between Greece and Turkey. Bulgarian+Turkish forces were actually redeployed towards the common border.
So, with a bit of misluck you may have WWIII breaking out because of the Greek-Turkish conflict.