How does one correctly punctuate a sentence that declares that one has a question? [duplicate]

I contend that the sentence needs slightly more editing than just the punctuation mark at the end.

Reword slightly, to leave the sentence as a declaration:

Another question I had was why people were swimming with the dolphins.

Avoid changing words, but make it clearer what the question was:

Another question I had was, "Why were people swimming with dolphins?"

The sentence your son wrote contains an embedded question:

why were people swimming with dolphins

that would end with a question mark were it written like this:

Another question I had was this: "Why were people swimming with dolphins?"

or this:

Another question I had was, "Why were people swimming with dolphins?"

Perhaps the teacher wanted one of those sentences, or perhaps she wanted the sentence to read thus:

"Another question I had was why people were swimming with dolphins."

The latter is an indirect question in normal indirect question word order. What your son wrote is also an acceptable (to me, at least, in informal spoken and written English) indirect question written in normal WH-question word order, which is probably what made it appear to the teacher that the entire sentence is a direct question. Either way it's written, a question mark is incorrect because the question is a subordinate clause in a longer declarative sentence.

Another question I had was why were people swimming with dolphins?

This seems wrong to me: it's incorrectly punctuated (the "?" is wrong). It's best to change the word order.

Another link to examples of embedded questions is this one.

The relevant information is this:

"Kate [a copy editor] moves on to the second sentence: The question is, how many re-readings are reasonable? Uncertain about how to treat a question ('how many re-readings are reasonable?') embedded in a sentence, she picks up [The Chicago Manual of Style] . . . [and] decides to apply the following conventions:

• The embedded question should be preceded by a comma.
• The first word of an embedded question is capitalized only when the question is long or has internal punctuation.
• A short informal embedded question begins with a lowercase letter.
• The question should not be in quotation marks because it is not a piece of dialogue.
• The question should end with a question mark because it is a direct question.
• Since the author has followed all these conventions, Kate changes nothing."

(Amy Einsohn, The Copyeditor's Handbook. Univ. of California Press, 2006)

Notice, however, that in the example with the copyeditor, the sentence is declared to be a question. In your son's sentence, a declarative sentence, he is merely informing the reader that he had a question. I think the teacher should have told your son that changing the word order would have made his sentence perfectly grammatical instead of ambiguous and questionable. Look at these two examples:

1. I asked her why people were swimming with dolphins.
2. The question is why people were swimming with dolphins.

versus

1. I asked her, "Why were people swimming with dolphins?"
2. The question is why were people swimming with dolphins?

The first two are indirect questions and the second two are direct questions expressed as dialog in the first and as a direct question in a sentence declared to be a question in the second. The word order is different. The first two take no question mark, but the second two do. English is a word-order language. Change the word order, and the rules change.

In any case, I find the teacher's response too simplistic and dictatorial. It's not as black-and-white as her statement makes it seem. She offers no explanations of how your son could have written the sentence so that it wouldn't need a question mark, or why she believes that it's a question rather than a declarative statement. In other words, she wasn't teaching anybody anything, only demanding that she be obeyed because she knows best.