Disclaimer: I'm not even English, so I very well can't be Rowling. Thusly and consequently, amazingly and shockingly, I do not own Harry Potter or any of the characters in it.
Remus had never been fond of nick names. He mostly blamed it on the utter and total lack of normal affinity out of the seclusion of his own home. Going to school, before Hogwarts, had been nothing but sitting quietly in one of the corners, listening to and viewing the lessons go by, recesses being spent reading.
Remus had also never been any good at saying no. whenever one of the classmates prone to ignoring him asked to borrow something (usually a spare pencil or eraser, which they could have quite easily acquired otherwise.) he could never find it in him to refuse, even considering a past full of unreturned pencils and erasers.
So Remus simply couldn't say "no" when they asked him if just Rems was okay, he could only blush a little at the gesture. But when they started calling him Moony, he found he had to do something about it. And so he did. He began calling them Prongs and Pad-Foot and Worm-Tail (for reasons as obvious as him being called Moony.), but instead of discouraging, it encouraged, and soon he wasn't even the only one using those names.
Now Remus is known around the school as Remus, Rems, Remmie, Rem, Moony, Moons and even the occasional Moon-myster and moony-kins.
Somehow, he had never discovered a fondness for nicknames. He didn't blame lack of kinship anymore.
And that was it. Hope you liked it. Have a nice day.