Jack London’s example, as usage in What does “not for nothing” mean?, is:
“Not for nothing had he been exposed to the pitiless struggles for life in the day of his cubhood, when his mother and he, alone and unaided, held their own and survived in the ferocious environment of the Wild.”
Note that this is quite distinct from the contemporary American idiom “not for nothing, but…”, discussed in Nick Nicholas’ answer to What does “not for nothing” mean?, and so overused by Aaron Sorkin in The West Wing.
In its literal reading, “not for no reason = for a very good reason”, it is understatement, litotes, of the kind you might expect of classicising or even classical authors. And indeed, Elyse Bruce’s Idiomation blog (Not For Nothing) traces it back through Robert Louis Stevenson, to Shakespeare’s Merchant of Venice (ii.5):
An they have conspired together, I will not say you shall see a masque; but if you do, then it was not for nothing that my nose fell a-bleeding on Black-Monday last at six o’clock i’ the morning,falling out that year on Ash-Wednesday was four year, in the afternoon.
She further identifies that Plautus used an equivalent phrase in Aulularia iv.3:
It was not for nothing that the raven was just now croaking on my left hand.
But the Latin is Non temere est quod corvos cantat mihi nunc ab laeva manu: “it is not random that…” So Bartlett’s Familiar Quotations, which she got Plautus’ quote from, was substituting a Latin expression of litotes with a familiar English expression of litotes.