It came upon a midnight clear,'That glorious song of old' is, in fact, the understood subject of the preceding clause. But it's extraposed (put outside) and the grammatical subject is 'it'. 'It' stands in here for a subject that is too long or (in this case) doesn't scan in the usual subject position.
That glorious song of old
Harsh, Paxo, very harsh.
UPDATE [23/12/12]: One of the twitterlinguists, @drswissmiss, pointed out the following:
@linguist_laura right dislocations are kind of special though, so the song is the subject in some ways and it is the co-referential pronounAnd she is, of course, correct. As I was typing the post I was thinking about the status of it - and one thing it isn't is an expletive. An expletive subject is something that English (among other languages) does when it just needs to fill in a gap. If you take a suitably seasonal verb like snow, then in Italian you can say nevica, 'snows', whereas in English you need to say 'it snows'. What snows? Nothing snows, and the Italian reflects that by not requiring a subject. English has this thing that it has to have a subject, so it bungs in this meaningless filler it. There does the same job in There's no easy way to say this, and you can check: it's not felicitous to ask 'Where's no easy way to say this?', so you're clearly not referring to a place when you say there.
— Mercedes D (@drswissmiss) December 21, 2012
But in the carol, something did come upon a midnight clear: the glorious song of old. So it is not an expletive it, it's a referential pronoun (i.e. a pronoun that refers to an actual thing rather than being a filler). That's why I called it extraposition in the original post. But that's not the full story. Because it does refer to the noun phrase 'that glorious song of old', we can say that it is co-referential with it (it refers to the same thing in the world). This is right-dislocation (putting some noun to the right of where it would normally be). When you do right-dislocation, you leave behind a pronoun to fulfil the grammatical requirements of the language (English's afore-mentioned need for a subject of some kind) and to point you in the right direction to get to the real subject.
You can also get left dislocation, which is pretty common. In some of the Romance languages it's called clitic left dislocation because the noun is moved to the front and a 'clitic' (a kind of pronoun which has certain special properties) is in the place where the noun 'should' be. Here's Spanish (example from Karlos Arregi's paper here):
Estos libros, Juan los leyó ayer.The object of the verb is estos libros 'these books', but it's fronted. This often has the effect of making the noun into a topic or the focus of the sentence. In English we usually do this to contrast with something else ('these books he read, but those other ones he hasn't started'), but in Romance languages it's used for more purposes. Some languages, known as topic-prominent languages and including many (or all?) sign languages, put the topic of every sentence at the beginning (usually without a clitic though) simply to indicate that it is the topic of the sentence.
These books, Juan read them yesterday (literally 'Juan them read').
Back to the carol. I don't actually know enough about the syntax of dislocation to say whether the dislocated noun phrase is 'really' the subject of the clause. There is some evidence that the pronoun and the noun phrase are connected in this type of construction: for instance, if the language has case inflections on nouns, they may have the same case ending. (Non-linguists, move on to the next sentence; for linguists, here's a thesis chapter arguing that the pronoun in Czech left dislocation is a spelled out copy of the dislocated element.) OK, non-linguists, back in the room.
To settle this question once and for all, I'm going to refer to playground logic:
'Antidisestablishment' is a very long word. How do you spell it?
Answer: I T.