To foreshadow a successful outcome, indicated by some circumstance or event.
As you might expect of someone who writes this stuff, I like crosswords. A clue that I came across recently was 'Soothsayer with a noisy implement'. Those of you who are familiar with the arcane rules of cryptic crosswords may have deduced that the answer to this is 'augur'. For those of you who aren't crossword aficionados, an augur is a fortune teller and an auger is a carpenter's tool - the 'noisy' keyword usually translates as 'sounds like' and clearly, 'augur' sounds like 'auger'.
To be more specific about augers and augurs, an auger is the 'bit' part of a carpenter's brace and bit, i.e. a drill. An augur was a Roman official with the job of predicting the future and advising on public policy. That might sound like a difficult task but, in practice, the augurs just had to look mysterious and feign the experience of receiving omens arising from the flight of birds or the appearance of the entrails of sacrificial victims, etc. - no doubt to the accompaniment of a good deal of toga flapping and rolling of eyes.
Of course, 'auguring well' has nothing to do with drilling neat holes but derives from the Roman augur's prediction of a good outcome as the consequence of some portent. Similarly, 'to augur badly' didn't mean 'to make an inaccurate prediction' but 'to predict a bad event'.
The phrase 'augur well' isn't a translation from Latin but originated, in the late 18th century, amongst the classically educated English elite. The first record of it that I have found is from a speech to the UK Parliament, given by the Duke of Richmond and published in the Parliamentary Register for 1778:
"I augur well from the readiness with which it [his request for papers about the movements of British forces] has been granted."
'Augur well' has much in common with 'bode well', which is also of ancient vintage and means much the same thing. A bode was a herald or messenger and was referred to thus as early as circa 888AD in King Alfred's Boethius De consolatione philosophiae. Like 'augur', 'bode' also had to wait until the 18th century to become absorbed into a common phrase. The first known use of 'bode well' comes from John Dryden's Works, circa 1700:
"Whatever now The omen proved, it boded well to you."