닳다 means "get worn out" 이름이 닳다 = means "name get worn out" "name fade out" It is a poetic expression. 입이 닳다 = 입에 침이 마르다 = 입이 마르다 = 혀가 닳다 means "tell over and over about something" ex) 나는 그에게 담배는 몸에 해롭다고 입이 닳도록 말했다 I have told him over and over again that smoking is bad for health. 괜찮아 그대 난 아파도 돼요 궂은 시련 손님이라 여겨 천 번 울고 단 한 번 웃는대도 그대 나처럼 같은 소원 빌 테니 온종일 썼다 지우죠. 그대 고운 이름 닳을까. 밤새 앓던 슬픈 약속 하나 두 눈을 떠도 꿈은 계속되죠 세상 어디든 그대 품만 같아 비틀거릴 때마다 오직 그대만 보여 괜찮아 그대 난 아파도 돼요 궂은 시련 손님이라 여겨요 천 번 울고 단 한 번 웃어도 똑같은 소원 비는 나의 그대 있으니 차디찬 바람 그대 한숨 같아 눈물은 삼켜내죠 혹시 따라 울까 봐 사랑해 그대 난 보낸 적 없죠 부디 혼자 가슴 치지는 마요 세상 뒤로 운명을 어겨도 그댈 위한 이 곳이 바로 나의 자리죠 먼 길 돌아 꼭 만나지겠죠 다음 생이라 해도 기다릴게요 It’s alright if I’m in pain. I’m a customer to bad heartbreak. Even if I smile once every thousand cries You’ll make the same wish as me All day, I write then erase. In case your pretty name gets worn out. There’s a sad promise that’s been aching all night. I open my eyes but the dream continues. Wherever I am in the world, feels like I’m in your arms. Every time I sway, I can only see you It’s alright if I’m in pain. I’m a customer to bad heartbreak. Even if I smile once every thousand cries. I have you, who makes the same wish as me. The cold wind feels like your sigh I’m swallowing my tears, in case they follow me I love you, I’ve never let you go. Don’t pound on your heart alone. Even if fate falls behind the world. This place for you is my spot as well After a long way, we will definitely meet again. Even if it’s in the next life, I will wait.

"이름이 닳다" means "name is worn out/ rubbed off" For example, when you see a name inscribed on a tombstone or a panel of memorial is worn out, you can say that. ex) 국립묘지에서 이름이 닳은 묘비를 보았다. At the National Cemetery, I saw a tombstone on which an engraved name was worn out. ex) 온종일 썼다 지우죠. 그대 고운 이름 닳을까. I have written your name but erased it over and over all day, because I am concerned that your beautiful name would be worn out. * Here is the link for a music video. "같은 소원" (Same Wish) sung by 천단비 (CHUN Dan Bi) from Korean Drama "대박"(The Royal Gambler) OST https://youtu.be/UicWdOLmENw?t=48s "입이 닳다" literally means "mouth is worn out / rubbed off (by using many times)" This phrase is used when a speaker express his/her frustration over repeating same words to a listener. Of course, there is no way to make your mouth worn out even if you talk same words repeatedly. Thus this phrase is a hyperbole, expression of using exaggerated words. ex) 또 지각을 했구나. 내가 지각하지 말라고 입이 닳도록 이야기했는데. You are late again. I told you over and over again that you must not be late. ex) 입이 닳도록 충고해도 소용이 없었다. It was of no use to advise over and over again.

어떤 책에 봤고 그리고 Twitter에도 본 적이 있었는데 오늘 노래 가사 속에 봤어요 "그대 고운 이름 닳을까 "

I've never heard '이름 닳다.' where did you see that expression?