For as long as i remember i have been called "kooter" it was a rl nickname from childhood (go ahead laugh i got over it long ago).

Over the years i have had to explain that it is simply a play on my real name and was given to me by friends and family. Hence why it became my in-game name.

However now i find it has caused me a great deal of grief. People are actually mean to me because they don't like my name. Because apparently someone uses that word in derogatory way.

Anyways riot is there anything i can do about this or am i out of luck? (I don't mind paying for a name change).

I don't think Riot will change your name unless they actually find it derogatory. If you feel your name is bothering others around you and that is going to effect how you play, then I'd suggest purchasing a name change!

Kooter is actually slang and derogatory, are you sure your friends weren't making fun of you without you realizing it?

In-game Summoner Name changes can be purchased from the shop

I'm a little confused here seeing as Kootz is your in-game name and that by itself shouldn't be a problem. It's hard to see where the problem would be unless you're literally referring to yourself as kooter to everyone you meet in-game, in which case, as Orcca mentioned, vanity name changes are available from the in-game store.