The song is called Senza un perché (Without a reason) and it is sung by the Italian singer Nada (whom I had never heard of before). Anyway, in the Young Pope series, the prime minister of Greenland gifts the song to Pius XIII when they meet in the Vatican (don't ask me why).
It's definitely not the world's greatest song or anything, but it has a way of staying in your head - for better or for worse.