Situation: Say I’m quoting the phrase, “Billy’s always been a good kid.” If I were writing in a formal situation, could I write the following sentence: "Everyone says that he “has always been a good kid”? (Assume I used the name “Billy” in the sentence preceding that one.)
If not, would writing, "Everyone says that “[he’s] always been a good kid” or using “[he has]” be my only options?
My reading is that, in that example, you’d have to write:
*Everyone says that he has “always been a good kid”? *
In other words, you can’t change the quoted words at all, otherwise you’d change them to suit your purpose. The other option, as you say, is to do your second, but that’s clumsier. At the end of the day, it’s up to you to change your intro to suit the quote.
If you are directly quoting, I don’t think there is any need to change it even in formal writing. After all, you are quoting someone else’s words; it’s not your own writing.
If you did want to edit such a quote for clarity, the way it needs to be done is the second one you suggest, using square brackets to indicate an editorial alteration of a direct quote:
If “Billy’s” means “Billy has” or “Billy is”, then it’s a contraction and not the possessive form, even though the possessive form “Billy’s” is spelled the same way.
As Giles says, “Billy’s” by itself is ambiguous. It could be a possessive, or it could be a contraction of “Billy has” or “Billy is.” The two latter are informal. From the context, it is a contraction of “Billy has.”
This sentence uses quotation marks to indicate a direct quotation, but your use of the word “that” to introduce it makes it an indirect quotation. If you simply omit the quotation marks, it will be perfect.
Also, I doubt that “everyone” has said the quoted phrase exactly as shown, so it’s not quite proper to use the direct quote anyway, in this specific example.