I think usually people tend to remember the part that is repeated the most. Meaning - in pop music it would be the chorus, in classical, for example, in Beethovens 5th symphony there's the ''destiny theme'' that goes through the part and changes a little time to time.
Also, judging on Eurovision, usually the winners are realllllly deep and sad or the other way around - happy and energetic. So... the lyrics are not everything - I personally always remember a song in Estonian but I don't know anything else apart one word in that language. So... if it comes to songs in foreign languages, the person will probably remember the part wher s/he heard a familiar word.
It probably is easier to remember music that is cheerful, for example, from advertisements. I personally can't get rid of ''kawaii crush'' song about those panda hats - and yet I'm a rock music fan.
Talking about rock music - it's probably easier to remember an awesome guitar solo. :3
And as for classic music, it's easier to remember a piece with something... unique, for example, for me it would me Tchaikowsky's Dance la Fae, Haydn's 94th symphony (the one with surprise) and pretty much every piece with clarinet solo goes fine with me. So.. I would like to say that people probably remember a classical music piece better if there's solo of and instrument that they can play.