Churchill thought that the invasion of Rhodes would drag turkey into the war on the Allies side. It's very debatable that it would have worked out that way, or that it would have been a good thing in general.

In the autumn of 1943 Churchill's disquiet over his inability to sway the U.S. chiefs of staff to the British point of view, and his increasing irritation with the flagging Italian campaign, led him to champion a British invasion of the German-held Greek island of Rhodes, the largest of the Dodecanese Islands, in the eastern Mediterranean. Churchill argued passionately that the boldness of such an operation would pay off by dragging Turkey into the war and setting the Balkans ablaze, thus enabling the Allies to threaten Germany via the "soft underbelly." The problem was more than a shortage of landing craft and the fact that such an operation violated the agreements reached at Quebec, which limited future Allied military operations in the Mediterranean. From the American viewpoint Churchill's enthusiasm for this scheme smacked of yet another excuse for evading or delaying Overlord.

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The Rhodes venture was typically Churchillian: bold, imaginative, and opportunistic, "an immense but fleeting opportunity," he called it, arguing that "it was intolerable that the enemy, pressed on all fronts, could be allowed to continue to pick up cheap prizes in the Aegean." It was also a prime example of Churchill's penchant for imposing his vast authority and intellect to overwhelm opposition and influence the outcome of events. As official British naval historian Stephen Roskill writes, Churchill's "hope of bringing Turkey into the war, which was the principal plank on which he rested his case, was an illusion; for the Turks could not have defended themselves as long as the Germans held Greece and most of the Aegean Islands."

Although he agreed with its basic premise, Brooke despaired that Churchill's dogged pursuit of the invasion of Rhodes would upset the delicate balance of U.S.-British relations. "The Americans are already desperately suspicious of him," wrote Brooke, "and this will make matters worse." True to form, at Cairo, Churchill refused to temper his insistence on invading Rhodes. "It got hotter and hotter," recalled Marshall. Finally the chief of staff had had enough and exploded: "[N]ot one American soldier is going to die on that goddamned beach," he thundered to the horrified assemblage.

Eisenhower: A Soldier's Life. Carlos D'Este (pp 465-466)

It probably also factored in that he wanted to protect British interests in the Mediterranean, specifically the Suez canal, as well as continue the fight in a theater that was under British Leadership. On the other hand you have to realize that in November, 1943 Operation Overlord was still a huge gamble. A failed invasion at Normandy would have been disastrous for the Allies. The night before D-Day was just as restless and worry-filled for Churchill as it was for FDR, Marshall, Eisenhower, and everyone else involved.