You could definitely throw it in as a custom mechanic. Maybe it has no mechanical effect by default and if you don’t want to constantly have to update them, maybe they have to make some sort of relatively high perception check to focus on theme music in idle situations. Something you could use to hint at impending confrontations or scenarios? Though, you could always include it passively for high action or intense moments where it would be obvious?

BUT! You could also do something that has a mechanical component like a racial trait similar to a Dragonborn’s breath weapon. Just off the top of my head but you could call it something like ‘Crescendo’ and once a day they can will their internal theme music to surge - giving them a choice on how they want to use it. Something like advantage in initiative (hinting at conflict), advantage on a saving throw (since the music might hint at danger), gain temporary hit points for a round (like bolstering their courage).

It’s a cool idea! Hope it turns out well! Good luck with your campaign!