Avoiding getting into any specific theoretical framework…
"Say" takes a mandatory complement, which the quotation is clearly able to fill:
- *Then they said.
- Then they said that the baby was born.
- Then they said something I didn't catch.
- Then they said, "What kind of a baby did she give birth to? Bring a candle."
We can throw more tests at it:
- They said "What kind of a baby did she give birth to? Bring a candle", in a strange voice. (Comes before adjuncts.)
- ?They said ominously, "What kind of a baby did she give birth to? Bring a candle." (It can come after light adjuncts, but sounds weird without the usual heavy-shift prosodic changes.)
- "What kind of a baby did she give birth to? Bring a candle," they said. (Topicalizes.)
- What did they say? (Can be wh-questioned.)
- What they said was, "What kind of a baby did she give birth to? Bring a candle." (Can be pseudo-clefted.)
- What they said in a strange voice was, "What kind of a baby did she give birth to? Bring a candle."
- *How they said "What kind of a baby did she give birth to? Bring a candle" was in a strange voice. (Cannot be stranded by pseudo-clefting, unlike the adjunct.)
- *They wanted to say, and say they did "What kind of a baby did she give birth to? Bring a candle." (Cannot be stranded by VP-preposing.)
- ??They said "What kind of a baby did she give birth to? Bring a candle," and we did it/so/the same with "Sez you." (Cannot be do-so contrasted.)
- They said "What kind of a baby did she give birth to? Bring a candle" in a strange voice and we did so/the same in an ironic voice. (Can be do-so elided.)
All of this shows that the quotation is a VP-complement (a "primary complement", in theories that distinguish things that way).1
So, what is it internally? From the first batch of examples, the verb "say" seems to want an NP or S (or a DP or CP, or whatever your theory uses) as its complement.2 And at first glance, it sure looks like an S, not an NP. But that can easily be shown to be illusory:
- They said, "The baby was born."
- They said, "The baby." (Looks like an NP.)
- I missed the start of it, but at the end, they said "was born". (Looks like a VP in some systems, but in, e.g., Chomskyan frameworks where the subject is a child of the inflection, this isn't any kind of valid component.)
- I missed the start of it, but at the end, they said "-by was born". (Clearly a fragment of an S that can't be parsed as-is.)
- They said, "No hablo ingles". (An S in Spanish, but not in English; surely the outer sentence is just as valid for monolingual English-speakers as for bilingual speakers, right?)
- They said, "No ah blue in glass?" (Nonsense that you can probably work out is a misquote of a Spanish S by someone who doesn't speak Spanish, but you surely don't parse it as an clause of the outer sentence.)
- They said, "Ekke Ekke Ekke Ekke Ptang Zoo Boing!" (Pure nonsense.)
So really, the quote acts as if it were a single word.3
Which doesn't answer the "NP or S" question. The way it's used seems more NP-like than S-like, but I can't make a good case for that.
Meanwhile, how can that quote be the head of an NP? Whether you want to say the quote has been "lexicalized" as an N, or that the head constraint is defeated here and instead we have an undifferentiated Utterance heading an NP, depends on your theory of choice—but it doesn't seem to be significantly different from any other case of weird things as objects.
The last case looks basically the same as the first:
- After a while the baby cried wawa waa waa waa waa waa waa.
- After a while, the boy said, "I was at the store."
The fact that you didn't put it in quotes is obviously just a matter of orthography. Pragmatically, it might imply that it's a paraphrase rather than an exact quote, but I think syntactically you're still offering a direct quotation (as opposed to an indirect quotation: After a while, the boy said he was at the store), so it should be analyzed the same way as the first example. (Also compare spoken "After a while, the boy was like 'I was at the store'", which, at least for some speakers, also pragmatically implies paraphrase while still acting like a direct quote.)
The middle case is very different; the quote is clearly not a complement. "One doesn’t usually birth in the daytime" is already a complete sentence without anything after it, and the tests immediately fail:
- *What did one doesn’t usually birth in the daytime?
- *What one doesn't usually birth in the daytime is "Bring a candle; light a candle".
In fact, you can't even attach an adjunct after the quote:
- It was dark; one doesn’t usually birth in the daytime; "Bring a candle; light a candle", under normal circumstances.
So, what is it? I think that depends on how you'd analyze the sentence without the quote. The quotation affects the first clause in a way pretty similarly to the second, after all.
One possibility is that the second clause is a parenthetical adjunct to the first. In that case, the quote could possibly be another parenthetical adjunct. (If not, it's parataxis, so move to the next paragraph, do not cross Go.) However you prefer to analyze parenthetical sentence adjuncts like the second clause, you can do the same with the quote (e.g.,
[S …s1… [AdvP [S s2]] [AdvP [Utterance quote]]]).
The other possibility is that you just have two independent sentences paratactically combined (and the implied causal relation is purely pragmatic). In that case, the quote is just another thing paratactically grafted on. Even if you want to treat parataxis as syntactic adjoining of like XP categories (
[S [S s1] [S s2]), that doesn't really make much difference (
[S [S s1] [S s2] [S [Utterance quote]]).
1. However, consider this: '"What kind of baby", they said, "did she give birth to? Bring a candle."' That seems valid. What happened? We're not topicalizing part of the quote (or topicalizing the whole thing and then focusing part of it), as can be seen with an embedding test: You can say "I was surprised when they walked into the room" and "I was surprised when into the room they walked", and you can say "I was surprised when they said, 'What kind of baby did she give birth to? Bring a candle.'", but you can't say *"I was surprised when 'What kind of baby', they said, 'did she give birth to? Bring a candle.'". You could try to argue from this that, say, quoting is actually a case of simple parataxis, which leaves the entire sentence not an actual S, which is why it can't be embedded. But that raises the question of why it can be embedded when the entire quote appears in normal position. Nor does it explain why we're allowed to build an invalid sentence with "say" missing its object in the first place, to then paratactically adjoin the quote. So, I think the answer is still that the quote is the direct object, and when you do normal syntactically-valid things to it they work in normal ways—but there's also some rule that lets you cut off a chunk of it and paratactically adjoin it to the first of the sentence (with the remainder still serving as the object), and, only if you exercise that option, you end up with a not-a-S that can't be embedded. But I wouldn't bet too much on that analysis.
2. I think with the right discourse setup you might be able to give it a PP or SC complement (that isn't acting surreptitiously as a quote), in which case (unless you like theories where such PP complements are actually S in disguise with lots of movement and deletion) things are more complicated, but I'm going to ignore that.
3. Of course you can (and presumably do, as a normal human listener) go inside the utterance and parse the quote. But the fact that the parse succeeds for "What kind of a baby did she give birth to? Bring a candle." and fails for "Ekke Ekke Ekke Ekke Ptang Zoo Boing!" seems to have no effect on our grammatical judgment of the outer sentence, or on what we can do with it. If you, as a grammatician, want to "subparse" the quote, you could invent some kind of bracketing to show how it fits inside the outer sentence, and I don't think anyone would complain.