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There have been a few times when a particular fanwork (usually a fanmix or vid) has permanently changed my associations with a song, and it fits so well that I can only think of it as being about a particular character/pairing/canon. For example:

Total Eclipse of the Heart (literal version): Revolutionary Girl Utena thanks to the vid Total Eclipse of the Heartsword
Girl With One Eye: Vriska from Homestuck via the fanmix Just As Horrible as Me (it's a Terezi/Vriska mix but I think of it as more of an Aradia/Vriska song)
Hide and Seek: Battlestar Gallactica via Meant Well: The Occupation despite TEN MILLION "mmm whatcha say" videos on tumblr
Cosmic Love: Puella Magi Madoka Magica (and Homura/Madoka in particular) from the vid Mahou Shoujo Requiem
Handlebars The Doctor, though I've seen it fit well for a number of characters.
Seamstress: Nikita&Alex from Nikita from the vid Seamstress

Does anyone else have any examples? Preferably where the lyrics really do fit surprisingly well. Or just songs which are remarkable fits where you thought of the connection independently.

If you don't do fanworks, I guess songs from movies/tv where it fit the scene PERFECTLY and you no longer hear it as being about the original subject. But songs written for the soundtrack don't count :)

EDIT: Hee, and here's a vid which matches an association I got from just hearing the song: Take me To Church to Lucrecia and Cesare from The Borgias. I haven't even watched the show but I have a friend who goes on about it a lot, and the song really is perfect.