We pronounce the name of the twenty-first letter of the alphabet homophonically with the word you.
Was this what the letter was always called (ever since the analogous letter in Latin), or did it at some point shift from being called [u] to [ju]?
According to the Oxford English Dictionary, "U" was not always pronounced with the initial /j/:
The name of the letter down to the 16th century was u , pronounced like the long u of French or Latin origin, and consequently undergoing the same change to /juː/ which took place in ordinary words. The completion of the change is indicated by the use of the letter (u or v ) to represent the personal pronoun you in such passages as Shakes. L.L.L. v. i. 60 and Dekker and Webster Westward Hoe ii. i. (Compare IOU.) In Scotland the name /u/ was locally in use as late as the 19th century.
The change in pronunciation of u is the result of changes in pronunciation over time:
The Middle English ū from French or Latin, on the other hand, has become the diphthong /juː/ , /jʊə/ , written u , ue , or u-e , as in huge , mute , future , cure , with reduction to /uː/ , /ʊə/ after s (= /ʃ/ , /ʒ/ ), j , and r , as in sure , jury , brute , rule , optionally after l , as in lute , lure , and more widely in American usage.
The the sound of U changed, and as a result the name of the letter itself followed this change in the 16th century. That is, U to represent the long u of French or Latin became /ju/, and because the name of the letter was related to this sound, the name changed as well.
Well, first of all, "U" and "V" weren't really distinguished as separate letters until the 17th or 18th century. So the name of the letter may not be any older than that.
If it is, though, I'd think it's related to the Great Vowel Shift. Same way as the name of the letter "A" must have originally been "Ah" and then turned into "Ei" (matching what we think of as the "long vowel"), the same would presumably have happened with "Oo" turning into "yu" in both pronunciation of words containing this vowel, and in the name of the vowel itself.
Some sounds aren't used in different languages. In Dutch it is pronounced [u]; it's a 'pure' sound, which isn't used in any English word I know. In German, however, it is pronounced as [ou] similar to "you", without the y.
As Alex points out, the pronunciation of the letter "A" must have evolved. But again, in Dutch, we pronounce it as [ah].
When exactly it was decided that it would be pronounced as such, I do not know.