I found an article where the author collected a few songs which people listen in '56 or were inspired by the events of the Revolution.

Ahogy lesz, úgy lesz
It's cover of Whatever Will Be, Will Be by Ray Evans. After the Revolution it was basically banned.
https://youtube.com/watch?v=7vL6igSQxPU [Embed]

Honvágy
Another cover: Memories Are Made of This - a hit of Dean Martin in 1956.
https://youtube.com/watch?v=5ruBE72iD8o [Embed]

Avanti Ragazzi di Buda
A march of the Italian football club Lazio, starts with a nod to the Revolution. Very political song.
https://youtube.com/watch?v=dbp4bjaiwjo [Embed]

See You Later Alligator by Bill Haley, was broadcasted frequently on Radio Free Europe.
https://youtube.com/watch?v=Ch1UQ47rWKU [Embed]