Book of Ruth

I am studying the Book of Ruth. In looking at the meanings of the names, I see that Mahlon meant "sickness" and Kilyon meant "End-of-the-line". Are these names that would have been given at birth? Or, did ancient Jews give an appropriate name as a child grew?

The book of Ruth tells us almost nothing about Machlon and Kilyon. We read that they were husbands of Ruth and Orpah and sons of Elimelech and Naomi and that's it. Their names in Hebrew, as you note, are remarkable. In Hebrew, Machlon invokes either " forgiveness ", or "sickness" and Kilyon is clearly derived from the Hebrew word for " destruction ." 

The Zohar prefers the former interpretation of Machlon as " forgiveness ." As the Zohar says, "Machlon was called so because the Holy Once Blessed Be He ultimately forgave him because he opposed his father’s plan and strove for justice. Kilyon was so called because he was wiped [ destroyed ] completely from the world…. So too, Ruth, the wife of Machlon, joined the Jewish people, so God forgave him so that his name is remembered. Orpah, the wife of Kilyon did not join the Jewish people, so God destroyed Kilyon and his name was not remembered in Israel" 

They were not given these names at birth. These are names that came later, just like Naomi changed her name to "Mara," as it says  “Call me Mara, because the Almighty has dealt quite bitterly with me." 

Zohar Chadash 71a