Hello,
I am trying to delete a character for the name on one account and renaming another character in another account. When I deleted it and tried to rename it says that the name was taken. The character that is being deleted is level 10.
Hello @Urchin;
Welcome to the Lost Ark Forums! 
In this case, when you delete a character you will need to wait for 60 days for the name to be available so that you can use it on another character.
I apologize for the inconvenience and thank you for the understanding!