From BBC news Magazine:
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The new Collins "hey-ho" entry, which recognises both the hey-ho and heigh-ho spelling variations, defines it as "an exclamation of weariness, disappointment, surprise, or happiness".
Heigh-ho, heigh-ho, it's home from work we go
Seven dwarves in Snow White
"It's the verbal equivalent of a shrug," says Duncan Black, an editor for the dictionary. "You say 'hey-ho' or 'that's the way it goes' or 'c'est la vie.'" It's quite a British way to stoically say "mustn't grumble", he adds.
The saying first appeared in print in 1471, according to the Oxford English Dictionary, which says it has nautical origins, meant to mark the rhythm of movement in heaving or hauling.
Eventually, it blended meanings with the similarly spelled "heigh-ho," which was first recorded in 1553 and is defined as an expression of "yawning, sighing, languor, weariness, disappointment."