Say it's Monday, June 9 today, and yesterday Paul told me that he will have a test on Friday, June 13. Are the following sentences correct, or should it be the other way around?
Paul said, "I'll have a test next Friday."
Paul said that he'd have a test the next Friday.
Besides, how should I refer to the Friday in the next week (June 20)?
Your use of them is correct.
Local dialects aside, "next Friday" is prospective (looking forward). It's what I would use right now to talk about the Friday a few days from now. By contrast, "the next Friday" is retrospective (looking back). You would use it in past tense, to refer to a Friday that is already in the past.
A clearer example:
Paul said, "I can't come to the party, I have a test next Friday."
Several months later, Paul heard Alice and Bob talking about how fun the party was and sadly said, "I couldn't go to the party, I had a test the next Friday."
Referring to the Friday after next Friday is largely a matter of dialect.
"The Friday after next" is the most common, and is prospective (like "next Friday"). The retrospective variation is technically "the Friday after the next", but I wouldn't use that; it sounds very stilted.
An alternative is "Friday week".
I have lived in different parts of Australia, and have noticed that people say different things: (writing this on Monday 9th) 'this Friday' (13th) and 'next Friday' (20th) or 'next Friday' (13th) and 'the Friday after (next)' or 'the following Friday' (20th). First solution: say 'Friday this week' or 'Friday next week'. Second solution: say the date.
Paul said, "I'll have a test next Friday."
The use of next Friday is fine here, although it's not entirely clear that the test will be on the 13th. I would assume the test could be on the 13th, or a week later on the 20th. In English, "Next Friday" is an unfortunate idiom, in that it can be used to mean "this upcoming Friday" and "the Friday after this upcoming Friday."1
What is odd, though, is Paul's use of "I'll have a test." We don't usually use the phrase will have when it comes to future test-taking. If Paul was a native speaker, I'd expect to one of these instead:
I have a test next Friday.
I'll take a test next Friday.
Paul said that he'd have a test the next Friday.
Again, I wouldn't expect to see this from a native speaker, because we don't usually use the word the before next Friday. Instead, I'd expect one of these:
if the test was on the 13th:
Paul said that he'd have a test next Friday.
Paul said that he'd have a test on Friday.
Paul said that he'd have a test this Friday.or, if the test was on the 20th:
Paul said that he'd have a test (on) the Friday after next.2
Paul said that he'd have a test next Friday.
Yes, "next Friday" has been listed for both, because it's used to describe both cases. After hearing a statement like, "Paul will take his test next Friday," it is very common for native speakers to ask for clarification, like this:
"Wait – do you mean Friday the 13th? Or next week, on the 20th?
In any case, I feel bad for Paul if his test is on Friday the 13th – that's bad luck!
1For more on this ambiguity, see some of the answers under this ELU question. Also, more about why this came to be can be found at this ELU question. This also seems to be a recurring question on ELU.
2Oddly enough, the preposition on here can be included or omitted.
Say it's Monday, June 9 today, and yesterday Paul told me that he will have a test on >Friday, June 13. Are the following sentences correct, or should it be the other way around?
Paul said, "I'll have a test next Friday."
Paul said that he'd have a test the next Friday.
There is no reason to insert a "the" before Friday in either the direct speech or reported speech.
Paul said, "I'll have a test next Friday."
Paul said that he'd have a test next Friday.
Besides, how should I refer to the Friday in the next week (June 20)?
The most generic way to say this is
Paul said that he'd have a test the following Friday.
Of course, you can alter the sentence to be more specific.
If the sentence was said on Saturday the 14th, you can say:
Paul said that he'd have a test yesterday.
However, this introduces ambiguity as to whether "yesterday" refers to the day of Paul's statement or the day of Paul's test. In general, however, the proximity of yesterday to test seems to lessen the ambiguity.
On Sunday the 15th through Friday the 20th:
Paul said that he'd have a test last Friday.
Beginning on Saturday the 21st, use the generic form, unless you want to start saying such things as "two Fridays ago."
The next Friday or Next Friday is practically nonmeaningful. I mean what's next doing there? If Friday comes, it'll be just Friday, why next? To be precise, you should call it this Friday which'll mean that you are talking about the Friday which is coming. Something like this week, this month or this year which means following whatsoever period.
Let's consider this case - Today is June 9 (Monday). So, I'll refer June 13 as This Friday. You'll find this good use of phrase on TV channels. They always advertise or notify the viewers that way (Say...The Matrix THIS FRIDAY on AXN) However, if you want to refer to Friday falling on June 20, then putting next in the same sentence makes sense. I'd call it this way:
My exam is on this Friday (13th) and Paul's on next Friday (20th) or as SydneyAustraliaESLTeacher (too big ;)) says - Friday next week or the following Friday (the Friday that follows this week's Friday in this case).
BUT whatever it is, to stay clear, always mention the date to avoid ambiguity especially if you are talking on a serious deal. :)
the Friday after next.