So I check out a piece of sheet music, song in key of C, and it has a chord labeled "F#dim." Since this chord can also be called "Cdim," why do they label it F#? Sure, it's got an F# note in it no matter what you call it, but it seems to me it would make more sense to call it Cdim when you're actually in the key of C.

I'm aware that each of the three diminished (diminished 7th?) chords can be named after any of the four notes that comprise it. Is there some convention or aspect of music theory that dictates which note is used for the name? Or is it just arbitrary personal preference that applies? Anyone know?