I own nothing.
Her Quenya father-name is Irissë, a name that would translate into Sindarin as Ireth. Straightforward enough, but even before the Elves of the hosts of Nolofinwë and his sons, and those of Findaráto and his brothers' followings, even before they begin adopting Sindarin names, Irissë is called by names not given to her by her parents.
When Irissë takes her niece Itarillë out of the camp in Mithrim to teach her how to shoot, and later, how to fight with a sword, their training together takes them close to a Sindarin settlement. Irissë possesses little ease in forming relationships with those outside of her family, but Itarillë grows to count the children of the settlement as her playmates and Irissë speaks with their parents while keeping a close eye on her niece. It is they, aunt and niece, who first lay the groundwork for lasting alliances between the Noldor of Nolofinwë's host, instead of simple "alliances of the hour" that fall apart as soon as the latest battle or skirmish is concluded.
For all that Irissë is ill at ease when having to speak with strangers (at least when having a conversation more involved than that of exchanging mere pleasantries), she finds that she gets along well with the Sindarin Elves, who don't pry into her personal life and are just as fond of Itarillë as she is. It is they who give her the two Sindarin names she comes to hold dear.
Aredhel Ar-Feiniel. Noble Elf, White Lady. At first Irissë can't tell if they're being serious when they call her that or not. She wonders if the Sinda who first calls her that isn't being tongue-in-cheek, but after a while, Irissë decides she doesn't care, and realizes, to her shock, that the Sindar are being completely serious when they call her that.
When the time comes for her and her family to adopt Sindarin names to use among the native Elves of Beleriand, Irissë uses Aredhel first, and Ar-Feiniel second, and gives no thought to calling herself Ireth. Her older brothers—not to mention her cousins, especially on Fëanáro's side of the family—tease are about it, but Irissë manages to let it slide off of her.
Pretentious? Maybe. But if there's one thing Irissë is sure of, it's that you need a bit of an ego just to keep up in her family. The Noble Elf rather likes her new name.