This is probably not right, but...
I *think* it has something to do with Gregorian chant. Modern musical notation started to develop around the 10th century. Naming notes using letters of the alphabet had been around since the ancient Greeks, but I'd have thought this sort of thing would have been standardised along with the rest of the notation. Solfege (do re mi etc to Julie Andrews' heart's content) was invented a lot later.

Rather than a system of musical keys, Gregorian chant used modes, i.e. scales that all used the white notes but had different notes as the root (most important note in the scale, usually the finishing note). Different modes give a different flavour to the music.

Each mode had an associated reciting note, the note that the priest stuck to when chanting important bits of the liturgy. This was generally five notes above the root (the dominant). The first mode in Gregorian chant was the Dorian, which has D as a root note (i.e. a scale consisting of all the white notes starting on D). The reciting note for that mode would be A. It would make sense, therefore, to assign the first letter of the alphabet to the reciting note of the first mode.

The above is pieced together from half-forgotten music A level lessons and a book of Hildegarde von Bingen tunes. I might, therefore, be talking out of my arse. Sorry.
(Concrete Cow, Wed 3 Jun 2009, 22:46, Reply)