Welcome, Mathilda!
Thanks for reaching out to the Apple Support Communities! If we understand correctly, you're running into a problem with changing the ringtone on your iPhone. We can understand wanting to make a change to your notification preferences. We will do all we can to help get you pointed in the right direction.
When you choose 'Pick a song', are you selecting a song you purchased from iTunes, one from Apple Music (if a subscriber), or music downloaded from another location? Have you tried to remove the song and re-add it to see if it corrects the issue? If not, you may want to start there.
Also, what type of iPhone do you use? Which version of the iOS software do you currently have installed on your device? You can find this in Settings > General > About.
Take care!