Gibbon begins with an account of his ancestors before moving on to his birth and education, which was partly private and partly at Westminster School. He matriculated as a student at Oxford University, an institution which he found at a low ebb.
To the university of Oxford I acknowledge no obligation; and she will as cheerfully renounce me for a son, as I am willing to disclaim her for a mother. I spent fourteen months at Magdalen College; they proved the fourteen months the most idle and unprofitable of my whole life.
Of one of his tutors Gibbon says that he "well remembered that he had a salary to receive, and only forgot that he had a duty to perform." Gibbon's father took alarm on learning that he had converted to Roman Catholicism and, in order to bring him back to the Protestant fold, sent him to live with a Calvinist minister in Lausanne. Gibbon made good use of his time in Switzerland, meeting Voltaire and other literary figures, and perfecting his command of the French language. He also fell in love with a Swiss girl, Suzanne Curchod, but his wish to marry her was implacably opposed by his father. "I sighed as a lover, I obeyed as a son." On returning to England he published his first work, the Essai sur l'ĂŠtude de la littĂŠrature. The next major event Gibbon mentions was his taking a commission in the Hampshire militia, an experience which he tells us was later to be of advantage to him:
The discipline and evolutions of a modern battalion gave me a clearer notion of the phalanx and the legion; and the captain of the Hampshire grenadiers (the reader may smile) has not been useless to the historian of the Roman empire.
He then details his travels through France and on to Lausanne, where he formed a friendship with John Holroyd, later Lord Sheffield, which was to last for the rest of his life. Gibbon crossed the Alps into Italy and eventually reached Rome. He had for some time wanted to begin writing a history, without being able to choose a subject, but now, he tells us, the exciting experience of walking in the footsteps of the heroes of antiquity gave him a new idea:
It was at Rome, on the 15th of October, 1764, as I sat musing amidst the ruins of the Capitol, while the bare-footed friars were singing vespers in the temple of Jupiter, that the idea of writing the decline and fall of the city first started to my mind.
After returning to England Gibbon engaged in several other literary exercises before finally beginning to write his Roman history. The Memoirs now give a detailed account of the years he spent producing its successive volumes, and of the many hostile criticisms his work attracted. These labours were diversified by his experiences as a Member of Parliament, and his writing, at the request of the Government, a "MĂŠmoire justificatif" asserting the justice of British hostilities against France at the time of the American Revolutionary War. During the course of writing the Decline and Fall Gibbon moved back to Lausanne. Gibbon's Memoirs end with a survey of the factors he considered had combined to bring him a happy and productive life.

Question: While in Rome, on what exact date did it first comme to Gibbon's mind to write about the Empire's decline & ultimate fall? 
Answer: October 15th 1764.
Alternative answer: October 5, 1764