Hi there,
yes,
Arwen is indeed a real Welsh name. It means muse. Maybe Tolkien didn't know that name already existed when he used it or he knew it anyway, we'll never know.
My sister is called
Arwen and when my parents chose the name they didn't even know about the LOTR character
Arwen.
My sis is very happy to have that name and she loves the LOTR.
Cwtch,
Megan