I read 涼宮ハルヒの憂鬱.
"ハルヒ" isn't a popular name for a Japanese woman, and "キョン" isn't a popular nickname, either. They are probably the author's own creation, so if you really want to get the correct answer, you should ask him.
Generally speaking, however, I can think of キョン as the nickname for:
恭一郎・京一郎・響一郎（kyou ichi rou) (man)
恭子・今日子・京子・杏子・響子 (kyou ko) (woman)
京太郎・姜太郎・京太朗 (kyou ta rou) (man)
恭介・恭輔・享佑・杏輔 (kyou suke) (man)
Any woman and man's real names that start with "kyou" can be called キョン as their nickname.
For example, 響一郎 is rather a long and "heavy" name. But キョン is a very "light" name. The man doesn't like the nickname, so he said, "This is an 'unwelcome twist' on my real name," I believe.
As for the difference between ん and ン, I don't agree with you.
”Kyon" can be written as either ”きょん” or "キョン."
きょン or キョん looks very weird to my eyes.
Although the answer here describes that be ん and not ン.
I don't think so.
One more thing to note:
小泉今日子 is a singer and actress in Japan, and her nickname is キョンキョン, not キョン.
Why キョンキョン, not キョン?
I think キョン is too "light." And it might resemble "チョン" which is a very dirty word in Japanese history.