I'm going from deep memory, so please don't hold me strictly accountable.
Once upon a time, up until WW II, enlisted ratings were divided between the more traditional naval trades and the more recent specialities. For the more traditional ratings (bo'sun's mate, gunner's mate, carpenter's mate, etc.), their badges were worn on the right arm, and they took pride in being "right arm rates". All others wore their badges on the left arm, and were presumably suitably humble about being inferior to the right arm rates. Nowadays, all ratings wear their badges on the left arm only. Where did it all originate? Probably with Jason and the Argonauts. Probably a long-ago budget-reduction initiative.
Best,
Hugh