Nyoro~n originated in a spinoff comic strip called "Nyoron☆Churuya-san" by Utsura Urarak, based on the character of Tsuruya from the anime "The Melancholy of Haruhi Suzumiya". The strip was later expanded into 13 two-minute episodes which are available on youtube by searching "Nyoro~n Churuya-san"; Watching one of these episodes is the best way to understand, through context, what the word means.
Nyoro~n is said in a cute, sad tone of voice after something unfortunate happens. The character who says it is a cute little girl who constantly fails to achieve what she sets out to do (often her goal is to get some smoked cheese).
People who use this word are probably either fans of the Haruhi series and its spinoff, or trying to look really cute, or both.
It gives all pennies.
Set out for a picnic.
It starts to rain.
You eat the food in the rain anyway.
You: Hello Best Friend, I got you some smoked cheese for your birthday!
Best Friend: I am lactose intolerant.
You (crestfallen): Nyoro~n...
You: Don't worry, I'll get you something even better, then!
Best Friend: But there is nothing better than smoked cheese.
You (realizing they are right): Nyoro~n...