In brief, Churchill became a PM because Lord Halifax did not want the job (at that particular time). The details of exactly why he did not want the job vary a bit from one account to another, but all accounts paint a very consistent picture.
- There is a lovely description of this story, with references, in the Wikipedia article on Lord Halifax. In particular:
As Lord Beaverbrook said, "Chamberlain wanted Halifax. Labour wanted Halifax. Sinclair wanted Halifax. The Lords wanted Halifax. The King wanted Halifax. And Halifax wanted Halifax." Only the last sentence was incorrect, however; Halifax did not want to become Prime Minister. He believed that Churchill's energy and leadership skills were superior to his own.
The given reference is:
Blake, Robert (1993). "How Churchill Became Prime Minister". In Blake, Robert B.; Louis, William Roger (eds.). Churchill. Oxford: Clarendon Press. pp. 264–270.
Blake's article says:

(The biographer above is The 2nd Earl of Birkenhead and the book is "Halifax: The Life of Lord Halifax", 1965; page 453. The 2nd Earl of Birkenhead served a Parliamentary Private Secretary to Lord Halifax in 1938–39 and as Lord-in-waiting to King George VI in 1938–40, so he also had an insider knowledge of the events.)
- A slightly different (but consistent with the above) take is given in
Roy Jenkins, "Churchill." London: Pan, 2002.
One important thing: Jenkins uses a variety of sources to support his description of the events (and is often critical of Churchill's own recollections).
In brief, Halifax had chickened out (of the PM job), but wanted to exercise indirect control on Churchill as a PM. The latter was a miscalculation.
This [Churchill's] account is not without a certain central truth, but is wholly inaccurate as to times and participants. The crucial meeting took place not on the Friday morning but on the Thursday afternoon; Margesson was also present; and it required no determination not to break a long silence on Churchill’s part to get Halifax to exclude himself. He had already done so at a 10.15 bilateral meeting with Chamberlain on the Thursday morning. There he stressed the great disadvantage he would suffer as a Prime Minister who was a peer, and for the first time used the phrase that the thought of being so ‘left me with a bad stomach-ache’.
The next day, Rab Butler, who was Halifax’s parliamentary undersecretary and very suspicious of Churchill, tried to get hold of his chief for a last-minute attempt to persuade him to change his mind. Butler was told that Halifax had gone to see his dentist, which suggested that his aches were not confined to his stomach, but also showed a high degree of disengagement. There is no doubt that Halifax was resolved not to become Prime Minister in the circumstances of 9–10 May 1940. Had he wished otherwise there is equally little doubt that he could have secured the appointment.
[The boldface font is mine.]
He was the preferred establishment candidate. The position of the Labour party was equivocal. This was key, for it was Chamberlain’s awareness that they would not serve under him which precipitated his decision to create a vacancy. Dalton was in favour of Halifax and, a little unreliably, reported Attlee as being so too. Morrison inclined that way. But, as Robert Blake has percipiently pointed out, they had ‘not a choice but a veto’. The veto they would not have exercised against Halifax, but nor were they inclined to do so against Churchill.
Throughout those two days of crucial political adjustment there is no doubt that, unlike Halifax, Churchill very much wanted to be Prime Minister. On the Thursday morning immediately following the conclusion of the debate and his late-night interview with Chamberlain he had telephoned Eden and asked him to lunch with him that day. When Eden got there he was surprised to discover a third figure in the shape of Sir Kingsley Wood, then Lord Privy Seal and hitherto a firm Chamberlain loyalist. But Wood had decided that, in Chamberlain’s own interests as well as in those of the country, it was time for him to go, and he was firmly convinced, even before Halifax had contracted his ‘stomach-ache’, that Churchill was the essential successor. His views were not crucial on that Thursday, for Chamberlain and Halifax had already come to that conclusion without Wood’s intervention.
What is more open to doubt is how self-abnegatory were Halifax’s motives. He was not called the ‘Holy Fox’ for nothing. He believed he was wiser if not more energetic or militarily focused than Churchill. He believed he might exercise more control over this impetuous figure (to whom, together with some of his associates such as Beaverbrook and Bracken, he was known to refer as ‘gangsters’) from a strong Cabinet position and with the dignity of having declined the premiership than if he were jostling with him for control of what would have been nominally (but only nominally) a Halifax government.
Furthermore the Churchill government might quickly fail. There were many who thought this at the beginning. In that event he (Halifax) could step in from a position of strength, unthreatened by a powerful rival, and pick up the pieces – although they would probably have been the fragments of an independent Britain as well as those of political power. Both these hypotheses were surmises, although there were a few hints in Halifax’s private correspondence that they had some validity. And both proved false. Over the remainder of 1940, until he was sent as ambassador to Washington, Halifax’s influence within the administration diminished rather than increased, and the Churchill government, so far from being ‘nasty, brutish and short’, survived for five years and became the most famous of the century, maybe indeed of the whole history of British Cabinets. For this we owe much to the fact that Halifax, who on 9 May could have become Prime Minister, wisely declined to do so.
...
Throughout those two days of crucial political adjustment there is no doubt that, unlike Halifax, Churchill very much wanted to be Prime Minister. On the Thursday morning immediately following the conclusion of the debate and his late-night interview with Chamberlain he had telephoned Eden and asked him to lunch with him that day. When Eden got there he was surprised to discover a third figure in the shape of Sir Kingsley Wood, then Lord Privy Seal and hitherto a firm Chamberlain loyalist. But Wood had decided that, in Chamberlain’s own interests as well as in those of the country, it was time for him to go, and he was firmly convinced, even before Halifax had contracted his ‘stomach-ache’, that Churchill was the essential successor. His views were not crucial on that Thursday, for Chamberlain and Halifax had already come to that conclusion without Wood’s intervention. They may have been more so on the next day when, following the early morning news of the German offensive in the West, Chamberlain was tempted to use this as an excuse for staying on ‘until the French battle was finished’. Wood’s signal service then was to tell him, as a friend.”
- Lastly, the following is taken from page 181 in
Jeremy Havardi, "The Greatest Briton: Essays on Winston Churchill's Life and Political Philosophy", Shepheard-Walwyn; 2010.
What is certain is that Halifax had decided not to accept the top job. The most glittering prize in British politics was there for the taking – but Halifax turned it down. Historians have long speculated on the
reasons for this spectacular self-denying ordinance.
In his diary,
Halifax wrote the following explanation:
I had no doubt at all in my own mind that for me to take it would
create a quite impossible position. Quite apart from Winston’s
qualities as compared with my own at this particular juncture, what
would in fact be my position? Winston would be running defense,
and in this connection one could not but remember how rapidly the
position had become impossible between Asquith and Lloyd George,
and I should have no access to the House of Commons. The
inevitable result would be that being outside both these vital points
of contact I should speedily become a more or less honorary Prime
Minister, living in a kind of twilight just outside the things that really
mattered.
[The italics are mine]