What were the names of Naomi's two sons?

Answer: Mahlon and Chilion

Mahlon and Chilion married Moabite women and died in Moab. Ruth 4:10 identifies Ruth as Mahlon's widow.

“The man’s name was Elimelech, his wife’s name was Naomi, and the names of his two sons were Mahlon and Chilion. They were Ephrathites from Bethlehem in Judah, and they entered the land of Moab and settled there.”— Ruth 1:2 (BSB)