Larrikin on the face of it would be a diminutive, meaning Little Larry; and as for who "Larry" would be, it sounds distinctly like a variant on "Harry", as in "Old Harry", meaning the Devil - which would suggest that calling "mischievous youths" larrikins would be equivalent to calling them young devils, which seems likely enough. After all, thats what they are called often enough to this day.

So perhaps "happy as Larry" is really "happy as the devil."