Green’s Dictionary of Slang

haw-haw n.¹

a dandy or an aristocrat, thus attrib.

1869		Maitland Mercury (NSW) 1 June 3/1: [H]e was a stranger, with a superfluity of jewellery, and a ‘haw haw’" style.
1879		Man about Town 11 Sept. 4/2: Bad temper [...] remorse, even despair are quite compatible with the glories of the scene; but no one is ‘hipped’—not even the haw-haws.
1874		Empire (Sydney) 19 Jan. 3/4: [T]he person in charge was a would-be heavy swell — a haw-haw sort of personage.
1880		Bulletin (Sydney) 6 Nov. 4/4: E’en though they walk upon their paws. / Apparently are tame. / Are sprucely dressed, and full of ‘haws!’ / They’re puppies all the same!
1906		H. Lawson ‘The Spirits of Our Fathers’ in Roderick (1967–9) II 189: And the tissue starts and wakens, summons ‘Haw-haws!’ to its aid.
1912		E. Pugh Harry The Cockney 191: I fair hate the sight of the people next door. A lot o’ superior haw-haws and niminy-piminy clothes-props!

In phrases

bit of haw-haw (n.)

a fop, a dandy.

1909

J. Ware Passing Eng. of the Victorian Era 30/2: When these young bits of haw-haw borrow a swallow tail coat and a crook stick, and a bit of window to shove into their weak peepers, and then go into the Gaiety with an order, strike us purple if they’re not at their best then. They know all the actresses of course, and the way they talk about some of ’em would make a red stinker turn blue.