Not just a song that can be found in the archives, but one that almost everyone can hum, even today.
(Somebody asked what was meant by "today's...." Throw whatever you want out, somebody tossed out "Love me tender" as being a tune from in the 1860s.)
As much as I want TiK ToK by Kesha to be a recognizable tune in half a millenia I know that's not happening. Personal Jesus by Depeche Mode is one of the most covered songs of the past 50 years so that very well may become immortalized through diffusion alone. There's a couple dozen jazz standards that could have that kind of staying power as well, especially considering their ubiquity in performance repitoires and books of sheet music.
I heard it on the radio recently and they censored the beginning:
Wake up in the morning feeling like [redacted]
??????? Why
P Diddy/Sean Combs has a lot of ongoing allegations right now
I heard a live performance where the line was changed to:
Bored music teacher in 2200: "and here children, we find the most important contributions to late 20th centure music: a phonograph of Depeche Mode's Violator."