tl:dr
The recruitment of civilian ships to evacuate the BEF was largely Ramsay's idea, set forth in the Dynamo Room at Dover during a conference on May 20th.
This is a bit difficult to answer.
One thing is clear. It was not Churchill studying the map the night before the operation Dynamo started then deciding this all alone on his own.
Responsible for the whole operation was Vice-Admiral Ramsay.
As Vice-Admiral Dover, Ramsay was responsible for the Dunkirk evacuation, codenamed Operation Dynamo. Working from the underground tunnels beneath Dover Castle, he and his staff worked for nine days straight to rescue troops trapped in France by the German forces. For his success in bringing home 338,226 British and allied soldiers from the beaches of Dunkirk, he was asked to personally report on the operation to King George VI and was made a Knight Commander of the Order of the Bath. (Wikipedia)
And also from Wikipedia Dunkirk evacuation
Without informing the French, the British began planning on 20 May for Operation Dynamo, the evacuation of the BEF. This planning was headed by Vice Admiral Bertram Ramsay at the naval headquarters below Dover Castle, from which he briefed Churchill as it was under way. Ships began gathering at Dover for the evacuation. On 20 May, the BEF sent Brigadier Gerald Whitfield to Dunkirk to start evacuating unnecessary personnel. Overwhelmed by what he later described as "a somewhat alarming movement towards Dunkirk by both officers and men", due to a shortage of food and water, he had to send many along without thoroughly checking their credentials. Even officers ordered to stay behind to aid the evacuation disappeared onto the boats.
It is however unfair to give credit to only one guy, when this really was a result of a large team work and a series of improvisations.
Plans to recruit or impress civilian ships were already in the drawers and ready to implement before the crisis unfolded. This was done under Ramsay's orders. But the speed at which the ports were lost meant that all those big transport ships already on the list of the admiralty were of little use for the evacuation from the shallow beaches.
Then Ramsay was contacted by Captain Wharton, head of the small vessels pool, who had already spotted 40 little ships available for coastal embarkment along the Thames on his own, without orders.
And again Captain Tennant landing at Dunkirk and being the senior naval officer there signalled back, to now:
Please send every available craft to the beaches east of Dunkirk immediately. Evacuation tomorrow night will be problematic.
And the public itself was also involved and needs to get some credit:
On May 26, 1940, the English realized they would need as many ships as possible to help the British Royal Navy rescue soldiers from Dunkirk. It was 40 miles by sea from Dunkirk in northern France to Dover in southern England. The trip usually took three hours. British officials were hoping that the Royal Navy would be able to move in to evacuate the troops efficiently. However, Dunkirk‘s beach was too shallow for large naval ships to navigate. Additionally, there were ships that had already tried to rescue troops and had been sunk in the harbor, blocking access for larger vessels. Admiral Ramsay, who was in charge of Operation Dynamo, realized they would need many small boats to carry the waiting soldiers to the large ships off shore.
On May 29, 1940, a BBC radio broadcast heard across England asked that anyone with a motorboat and/or experience with boats volunteer to help with the evacuation. In response to the BBC broadcast, ordinary citizens from throughout England gathered at set locations with their small boats and created convoys of little ships to cross the Channel to Dunkirk. Some of the ships used in the evacuation included: motor boats, yachts, fishing boats, ferries, tour boats, and tugboats. The owners and skippers of these boats were ordinary British citizens who were willing to face incredible danger to help their country.
(Jean McGinnis "Miracle at Dunkirk", 8th grade reading paper, 2011.)
This little ships gathering was only really begun a few days after Churchill gave in to General Gort and ordered the operation to begin:
The decision was to seem to have been the critical act responsible for saving the B.E.F. But not until the evening of May 26 was Auphan sent across to Dover to enquire into the Royal Navy's plans and to try to concert French operations with them. He and his party were astonished by the immensity of the preparations explained to them by Vice-Admiral Ramsay. […]
Moreover, he was deeply impressed by the bombardment Dunkirk was sustaining and concluded that the port was now useless. "Maritime road very precarious," he signalled Darlan that morning, May 28. "Withdrawal by sea of all important units seems to me impracticable."' Hence it was not until afternoon that a decision was taken at the top. Darlan ordered the assembling of an evacuation fleet 24 nine days after the Royal Navy had set to work, two days after Operation Dynamo was begun. And though a common action was at last in process, a time of disputes and recriminations was upon them.
Believing that Gort was unwilling to contribute to the last defence of the bridgehead, Abrial repeatedly asked Weygand to hold him to this task. At the height of the rescue operations, May 31, Darlan's headquarters signalled Odend'hal crankily that Dynamo was "an operation long prepared on the sly and which seems to be being carried out to the detriment of the good order necessary to the defence of the fortified camp of Dunkirk."
(From John C. Cairns: "De Gaulle Confronts the British: The Legacy of 1940", International Journal – Canada's Journal of Global Policy Analysis, Vol. 23, No. 2 page(s): 187-210, 1968.)
But since Churchill was now the prime minister, he gave the ultimate go-ahead:
Instead, as France lurched toward dishonorable surrender and puppet status under Marshal Pétain, Churchill concentrated on getting the British Expeditionary Force safely back home. And he succeeded. Nine-tenths were rescued from Dunkirk, and many Allied soldiers with them, more than three hundred thousand in all, brought back by an improvised armada of ships, great and small, including pleasure cruisers and fishing boats, which gave picturesque color and even romance to the story, a typically British tale of snatching victory from the jaws of defeat.
(Paul Johnson: "Churchill", Viking: New York, 2009, p 113.)
It should be noted that the movie presenting this claim not overly concerned with historically accurate details in general, and that the little ships did not play really such an important role in the evacuation itself, but in boosting morale (Nicholas Harman: "Dunkirk: The Necessary Myth", Coronet: London 1990).
Sources:
Christopher Hibbert: "Operation Dynamo", in: History of the Second World War, Part 6, 1972 p 159.
Walter Lord: "The Miracle of Dunkirk",Viking Press: New York, 1982, p 172. (p 42-46, & ch. 9, 154–175.)
