An extreme standard of comparison; for example, "It rained like billy-o; we were all soaked through.".
There isn't a clear-cut explanation of this phrase's origin. As it involves a common name, it is easy to come up with eponymous suggestions; for example, the 19th century steam locomotive the Puffing Billy. Alternatively, the derivation is said to be from Joseph Billio, the zealous 17th/18th century Puritan preacher. Billio preached at the United Reformed Church in Market Hill, Maldon, Essex, in and around 1696. He was an enthusiastic 'hellfire and damnation' preacher and, given his name and reputation, ought to be a serious contender as the source of the phrase. They are certainly convinced in Maldon, and it must be true - they have a plaque to prove it.
Regrettably, the phrase didn't become common until long after Billio's death and disappearance into obscurity (had you heard of him before?). The earliest printed record of the phrase in print is in the US newspaper The Fort Wayne Daily Gazette, March 1882:
"He lay on his side for about two hours, roaring like billy-hoo with the pain, as weak as a mouse."
Around the same time, a similar phrase emerged in the USA. In the North Dakota newspaper The Bismarck Tribune, September 1883, there's a piece of what Mel Brooks would have called "authentic frontier gibberish", under the title An Old Frontiersman Talks:
"You say Ol Grant was 'yar with the gang
And the capital's one hoo-doo
And the people cheered him like billy-be dang
Why, pardner, it can't be true!"
This latter example appears to be a version of 'Billy-be damned', which had been recorded earlier, in 1849, in the Gold Rush Diary:
"Theodor looked as big as billy-be damned"
Those citations both have the ring of minced oaths, that is, euphemistic phrases that endeavor to avoid speaking the name of religious figures. It seems probable that 'billy-be damned' and 'billy-o' were references to the Devil. 1849 sees the first use in the USA (and possibly anywhere) of the name billy-goat to denote a male goat. The term 'billy-be damned' could well have been used to indicate the Devil, following the imagery that pictured Satan as a monster in goat's form.