According to Phrase finder:
There are two commonly espoused contenders. One is the Australian boxer Larry Foley (1847 - 1917). Foley was a successful pugilist who never lost a fight. He retired at 32 and collected a purse of £1,000 for his final fight. So, we can expect that he was known to be happy with his lot in the 1870s - just when the phrase is first cited.
The alternative explanation is that it relates to the Cornish and later Australian/New Zealand slang term 'larrikin', meaning a rough type or hooligan, i.e. one predisposed to larking about. 'Larrikin' would have been a term that Meredith would have known. The earliest citation of that is also from New Zealand and also around the time of the first citation, in H. W. Harper's Letters from New Zealand, 1868:
"We are beset with larrikins, who lurk about in the darkness and deliver every sort of attack on the walls and roof with stones and sticks."