In my opinion, the forename Larry was simply chosen as a jocular reinforcement, a variant reduplication, of the adjective happy, the first syllable of each word containing the vowel-sound /a/, the second syllable the vowel-sound /i/.
The earliest occurrences of (as) happy as Larry that I have found are from Australian publications, but seem to indicate that the phrase may have originated in Irish-English.
The earliest instance that I have found is uttered by a fictional Irishman in The Murray Land Bill, No. 2., by a person signing themself G. U. A., published in The Illawarra Mercury (Wollongong, New South Wales) of Monday 23rd November 1857:
Mark the poor man’s progress. He reads and dreams of the bright, sunny land of Australia; of its millions of acres of waste land only wanting to be occupied, cleared, and cultivated, to make him rich and independent,— that is comparatively so; he dreams glad dreams of peace and plenty for self and family, and, despite his strong love for the land of his birth, even for his own smoke-colored mud edifice, and the pig in the parlor, he makes up his mind to “go emigratin,” and in due course finds himself on the wide waste of waters, an adventurer for the good things of Australia, where he his [sic] landed “with a light heart and thin pair of breeches,” ready and willing for anything, looking forward to the “bit o’ Iand” which is to make himself, and Judy, and the gorsoons, both boys and girls, gintlemin for life. “Begorra, Judy, asthore, we’ll be all gintlemin intirely when we get to Austhralia, so don’t be lettin on so about lavin the ould country; put down your praskeen, ma collendhas, an don’t be cryen the eyes out o’ yer head; look at little Molly there roullen about the floor, sure we’ll make a lady of her; an, with the blessing of God, we’ll send for the ould father and mother, an we’ll all live together, like Brown’s cows, and be as happy as Larry.” So argues Pat, and no doubt so argues John Bull and Sandy.
The second-earliest occurrence of happy as Larry that I have found is from Bill Smith. The lay of an engine driver, by one Nahum Dodge, published in The Irish Harp and Farmers’ Herald (Adelaide, South Australia) of Saturday 3rd February 1872:
Ole Bill Smith was the engineer
Of a boat called the Sandridge Belle—
A rale “dugout,” but a good un to go—
One New Year’s Day he was steamin’ along,
Full of passengers out for the day;
Bill was happy as Larry, a singin’ a song,
And keepin’ her full and away.
Another early instance is from Mount Bischoff Notes, by ‘Flaneur’, published in the Launceston Examiner (Launceston, Tasmania) of Monday 28th may 1883:
An old couple are as happy as Larry. They have just tied the nuptial knot. Widow Mackree is sick of stirring the fire all alone, and Barney O’C. is following suit with the rabbits and hares. Young single men, what are you frightened of when a bridegroom of 77 years and a bride of comparative age have experienced the blessings bestowed by Hymen, and would not even allow the few more years nominally allotted to them to pass away like hatters?
The way so many self-styled ‘etymologists’ make up complicated stories in order to explain simple phrases is fascinating. For instance, to rain cats and dogs—a phrase simply based on a cat-and-dog fight as a metaphor for a storm or hard rain—is popularly explained as referring to the (uncorroborated) fact that heavy rain would occasionally carry along the streets dead cats and dogs! Likewise, since the 19th century, several persons have taken leave of their common sense and made up theories purporting to give rational, historical origins to the idiom to fight like Kilkenny cats—which originated in the tale of two cats fighting until only their tails remained, a tale meant to be nothing but amusing nonsense.
In the case of happy as Larry, one of the stories that have been made up, without a shred of evidence, is that the phrase originally referred to the Australian boxer Laurence ‘Larry’ Foley (1849-1917); it is interesting to remark that the earliest instance of the phrase that I have found dates from 1857, when Foley was about eight years old…
According to another of these stories, the last element of happy as Larry refers to larrikin, a noun used in Australia and New Zealand to denote a street rowdy—an equivalent of hooligan. But, if this were true, why has the intermediate form happy as a larrikin never been recorded?