In which year did Dumbledore become the Headmaster for Hogwarts school of Witchcraft and Wizardry?
Although the generally accepted date is 1956 (e.g. presuming that Minerva McGonagall replaced Dumbledore as Transfiguration Mistress on his appointment to Headmaster) there's a fair argument that he didn't actually become Headmaster until late 1964/early 1965.
We know from the latest release on Pottermore that Minerva McGonagall was appointed as Dumbledore's assistant in 1956 but not as his replacement. There's no canon description of how long she worked under him but we can reasonably assume it would take quite a few years to go from a 21-year-old trainee to full Mistress at Hogwarts;
She sent an owl to Hogwarts, asking whether she might be considered for a teaching post. The owl returned within hours, offering her a job in the Transfiguration department, under Head of Department, Albus Dumbledore. [Pottermore website]
Lupin was born in 1960 and didn't become infected until 1963/4. He speaks of the change in attitude at the school when Dumbledore became Head.
I was a very small boy when I received the bite. My parents tried everything  It seemed impossible that I would be able to come to Hogwarts. Other parents weren't likely to want their children exposed to me. But then Dumbledore became Headmaster, and he was sympathetic. He said that, as long as we took certain precautions, there was no reason I shouldn't come to school. Prisoner of Azkhaban
You can see an extensive discussion of this alternate hypothesis here.
Though never specifically stated, we can infer that Dumbledore became Headmaster in 1956. How do we know this?
Prior to being Headmaster, he served as Transfiguration professor under Headmaster Armando Dippet. Thus, when Dumbledore succeeded him he would have left a vacancy for the spot of Transfiguration. We know that Minerva McGonagall filled this vacancy, but the question is when?
From Order of the Phoenix, Chapter 15: The Hogwarts High Inquisitor:
"How long have you been teaching at Hogwarts?" Professor Umbridge asked.
"Thirty-nine years this December," said Professor McGonagall brusquely, snapping her bag shut.
Order of the Phoenix takes place in 1995 and 1996, with this statement coming in approximately September 1995.
Given the odd date of hiring (in the middle of the school year), we can infer that Dippet died in office, Dumbledore succeeded him, and McGonagall hired to replace him as soon as possible.
Additional Evidence for 1956
We also know that Tom Riddle jynxed the position of Defense Against the Dark Arts when Dumbledore (as Headmaster) refused him the position. Again, this is another date that is unconfirmed but is generally accepted as circa 1957.
Unfortunately this is based on the ambiguous
ten years after the death of Hepzibah Smith (which occurred an unspecified number of years after Riddle graduated), and on the above evidence pointing to 1956.