Her parents named her Lúthien, after the flowers that grow in their kingdom. Tinúviel she acquired later, after her singing was overheard. Which is... fine. Not what she would have chosen for herself, but fine. Like the flower she blooms for her own sake and like the nightingale she sings because she wishes to. What others get from that is their own business.
Still, when the court becomes boring and fatherly attentions press to heavily, she often wanders alone along the borders of Doriath, tracing sections of the enchantment her mother Melian wove. Sometimes she thinks she might like to pass through one day, to see the world beyond through more than story, feel the weight of the enemy's press herself. To feel something other than gentle shelter, see what she might achieve through her own powers.
She dances alone, expressing the feelings she is unwilling to put to song, to chance that someone might hear.