Please tell me which one is correct...
They said "no" because the children needed an adult with them.
OR
They said, "No," because the children needed an adult with them.
OR
They said no because the children needed an adult with them.
OR
Some other way?
Please advise - thank you so much!
If you are writing in the United States, you may be interested to learn that many readers are accustomed to seeing yes and no handled in what the Chicago Manual of Style, fifteenth edition (2003) calls "the ordinary way." Following is that guide's brief discussion of how to style yes and no (part of a very long section 5.202, headed "Good usage versus common usage"):
affirmative, in the; in the negative. These are slightly pompous ways of saying yes and no. They result in part because people are unsure how to punctuate yes and no. The ordinary way is this: he said yes (without quotation marks about yes, and without a capital); she said no (ditto).
Because Chicago presents all of its in-text examples in italics, it had to run yes and no in the examples above in roman (non-italic) type, but in actual practice "he/she said" would be in roman and yes or no would be in italics. Consequently, for the original example you give, Chicago recommends the following treatment:
They said no because the children needed an adult with them.
You have many other style options to choose from, of course, and you are free to follow your own preferences unless you're required to conform to a house or institutional style created by someone else; but I think that the Chicago recommendation in this instance is both clean and easy to understand, which makes it (in my view) an appealing choice.
If you're thinking of dialogue in a story, it makes sense to use your second formulation.
If you're just trying to point out that they denied the request or whatever it was that preceded them saying no, then your third formulation makes sense.
I don't think your first would ever be appropriate.
Punctuation is intended to help, not present additional problems. In spite of traditionalists' reactionary views on maintaining the rules they were once taught as sacrosanct, modern usage is increasingly disregarding them where it is considered reasonably safe and advantageous to do so.
Many style guides now accept an introductory colon or zero punctuation in place of the once unassailable introductory comma before a quote.
BusinessWriting gives examples of zero punctuation before quotes that are not direct speech:
The poem is titled "Ode to the Semicolon."
This is the last line of Frost’s "The Road Not Taken": "And that has made all the difference."
NUI Galway Marketing and Communications Office Style Guide includes an example using direct speech:
The inverted commas and capital letter are obviously sufficient markers, and the introductory stop can be chosen to indicate the writer's preferred intonation instead.
I'd be quite happy to extend this practice to terminal punctuation, and I'm fairly sure some authors do. Notice the difference between:
(1) I said "No" because I think that's the correct answer here.
and
(2) I said "No", because I realised just what demands on my time it would involve.
In (1), the focus is on the choice of word. In (2), the focus is on the refusal.
If it is not too important to show that the words are quoted verbatim, the softer 'quasi-quote structure' seems better. I agree with Sven that CMOS has a good suggestion here.
