See:-
http://www.winstonchurchill.org/i4a/pages/index.cfm?pageid=434
By the time Churchill resigned on 4 April 1955, it had been determined
that no further dukedoms would be offered except to Royal personages.
Yet WSC was different from other Prime Ministers and an exception was
considered. The Palace asked Colville if they could offer a dukedom,
confident that Churchill would refuse it. Colville took some
soundings. Churchill told him that he would never accept: "First of
all what could he be Duke of?" Colville reported. "Secondly, even if
he were Duke of Westerham, what would Randolph be? He could only be
Marquess of Puddleduck Lane which was the only other possession he had
apart from Chartwell. And thirdly, and quite seriously, he wished to
die in the House of Commons as Winston Churchill."
The oddest thing then happened. On April 5th the PM donned his frock
coat and top hat for his Audience, and Colville, knowing he was
hopelessly in love with The Queen, feared that despite all WSC’s
assurances he might accept out of his affection for her!
Churchill returned from the Palace with tears in his eyes: ‘Do you
know, the most remarkable thing—she offered me a Duke." With
trepidation Jock asked what he had replied. "Well, you know, I very
nearly accepted, I was so moved by her beauty and her charm and the
kindness with which she made this offer, that for a moment I thought
of accepting. But finally I remembered that I must die as I have
always been—Winston Churchill. And so I asked her to forgive my not
accepting it. And do you know, it’s an odd thing, but she seemed
almost relieved."