For the behind-the-scenes discussions on upgrading the Duke of Edinburgh's title that were mentioned by meeralakshmi, beginning with Queen Elizabeth II's suggestion in 1954 that her husband be created "Prince of the Commonwealth", see here:
At first he was 'just' the Duke of Edinburgh, making him a prince of the realm was his upgrade.
The correspondence in the above link also discusses the inconsistencies and disagreements over whether Philip was already a Prince or not (for example, the 1948 letters patent on the titles of his children referred to him as His Royal Highness Prince Philip, but there was a draft version of the same letters patent in which King George VI specifically edited out the "Prince").
Some pointed out that regardless of the official debate, he was already called Prince by the general public:
Evening Standard article, February 8, 1957:
No one has dared to clarify the situation, but surely the Duke's, and presumably the Queen's, view was made clear in 1954, when a Royal Society of Arts medal, which he chose, was struck with the inscription "Prince Philip, President."
Thus the Duke, who signs as Philip, the prerogative of a Prince, has vindicated the name, by which the man in the street has always called him.
Draft press release from the Prime Minister's Office, 1957:
Draft Guidance to Press
The Prime Minister and his senior colleagues felt that the great services which the Duke of Edinburgh has rendered to the country and his unique contribution to the life of the Commonwealth, culminating in the tour which he has just concluded, ought to receive some significant mark of recognition.
They therefore proposed to The Queen that the Duke of Edinburgh should formally be given the title and dignity of a Prince, and The Queen was graciously pleased to approve this proposal. In giving effect to it The Queen has let it be known that she would like the Duke of Edinburgh henceforth to be known as His Royal Highness The Prince Philip, Duke of Edinburgh, thus confirming a title which common usage and general affection have already largely accorded to him.
The effect of the announcement is really to give His Royal Highness the same position and title that he would have if he had been born a member of the Royal Family. It does not, of course, affect the succession. But it does signify appreciation of the unselfish devotion of His Royal Highness to the public interest.