I was just re-watching the anime and it seems like Kyon is only a nickname. Are we ever told what his real name is? Maybe from the light novel? And how come Kyon has such a nickname? Does it have any meaning?
No, we don't.
From quick googling, it seems that it has not been revealed yet, even in the light novel. But it seems some hints exist.
From "Kyon" on Wikipedia:
The name Kyon is actually a nickname given to him; his real name has yet to be revealed in the series. His sister is responsible for spreading its use amongst his schoolmates, much to his embarrassment and annoyance, and he misses being called Onii-chan (お兄ちゃん, Lit. "older brother") by his sister. In the ninth volume of the light novel (The Dissociation of Haruhi Suzumiya), his real name is hinted at in a conversation between himself and Sasaki, a girl who had known Kyon since middle school, who said that it was a very majestic name that didn't fit him.
His sister's name also hasn't been revealed yet.
Thread in MyAnimeList discussing this - there are many hints and guesses being discussed there.
All we KNOW about his name is that it's majestic/noble. As stated before, Sasaki, someone who knew Kyon before, stated this, but still referred to him as Kyon.
As well, based off what was said before, his last name could start with "Sa," or "Shi," assuming the second mora of his family name is after "Ka" so that he is seated behind Sakanaka Yoshimi. Based off Haruhi's last name "Su"zumiya, it's safe to assume that his last name also begins with "S"
On those two points, anything we conclude must fit under those two points. Though John Smith fits under the second, meaning his name in Japan would be Smith John, it does not 'fit' the first one.
Haruhi, also wouldn't know what his name is right away, as EVERYONE including the teachers refer to him as "Kyon", this means that "Kyon" must in some way be related to his actual name.
A quick look up on Bing, says Kyon is a ancient Greek word, meaning 'Dog', which certainly fits his dog-like demeanor towards Haruhi.
Speculation Based off all this info, as his name must be related to "Kyon", his name must be majestic/noble, and his family name begins with "S" it's possible his family may be Shimazu, or a similar Sengoku-era family name.
Speclation As well, his given name might simply be Kyo.
There are a lot of conversations and speculation about what his name actually is, and the reason being... we are never told what it is. The nickname Kyon came from his aunt, and his sister caused for the nickname to spread to everyone else, much to his chagrin.
If I remember correctly, logical deduction leads us to know that his last name starts with an "S", based upon where he sits at the beginning of their freshman year of high school. However, everything else beyond that is speculation.
The only other place his name gets mentioned in any detail is the spot you mentioned in your original question: when Sasaki makes a comment about it.
Well, I will provide some information for ya...
In the light novel (chapter 10b), Tsuruya called him "Kyorosuke", and if we replace "r" with "n", Voilà! his real name is "Kyonosuke". The reason why he's being nicknamed as "Kyon" is because of his parents' economical problems.
I saw someone's comment at haruhisuzumiya.net about this...
Kyorosuke is a nickname for Kyonosuke. Tsuruya just replaced “yono” with “yoro”, just like most of her statements with nyoro
My own thoughts
There is a theory that John Smith is his real name, but I don't think it really is. Yes, his name is not mentioned anywhere yet. Kyon's sister is also "Kyon's sister" for now ^_^. Not revealing name also may hint that he plays a more important role in universe than we think. He must be some kind of Kami or something <_<