That is very peculiar. I’ve never heard a sound that could be described as a rattle coming through the headphone jack. Does it sound more like a mechanical rattling noise or an electrical oscillation? An oscillation could sound like anything from a
“thump thump” to a high pitched whine.
Since the sound is coming from the headphone jack, I’m suspecting the circuit around the LM386 audio amplifier chip could be the issue. Please double check (and even re-solder) the pads in the audio path…R14-16, R19-20, R22, C8-11, C13, C3-4, U3, and of course make sure the speaker and the headphone jack are properly soldered.
Good luck.
-Brach