Green’s Dictionary of Slang

1. one’s concern, a problem.

1901		A. Binstead More Gal’s Gossip 129: ‘For so peautifil a madchen as yourselluf, ma tear, vhere shall I find a suitable barty?’ ‘That’s your piece-of-pigeon, not mine,’ replied the beautiful young girl.
1918		Sun. Times (Perth) 3 Mar. 20/2: ‘Well,’ summarised one of them gloomily in conclusion, ‘it’s his own pigeon.’ ‘His own pigeon,’ endorsed the other solemnly.
1921		D.L. Sayers Unpleasantness at the Bellona Club (1977) 149: Dunno. That’s your pigeon.
1935		G. Ingram Cockney Cavalcade 206: Well, it’s your pigeon.
1936		Queensland Country Life 27 Feb. 8/6: That we do not place the same construction upon them is our own pigeon.
1955		D. Niland Shiralee 89: It’s my pigeon [...] Let me worry about it.
1961		H. Livings Stop it, Whoever You Are (1962) Act I: It’s all his pigeon, you know.
1968		A. Salkey Late Emancipation of Jerry Stover (1982) 206: Besides, it’s Rybik’s pigeon, not mine.
1979		J. Morrow Confessions of Proinsias O’Toole 128: That’s Punchy’s pigeon.

2. one’s choice or preference.

1896		Eve. News (Sydney) 19 Oct. 3/5: The arrangement of the programme is his own particular pigeon, and his draft scheme is always submitted to the council of the association.
1939		Cumberland Argus (Paramatta, NSW) 1 June 13/5: ‘The bridge is too narrow for two cars to pass. If they try to pass on the bridge, it’s their own pigeon,’ said the engineer.
1943		R. Llewellyn None But the Lonely Heart 38: Darts was never His Pigeon so He always give it a miss.
1950		Wkly Times Melbourne 17 May 333/4: If prisoners take to religion / And turn with a smirk to the East, / That is, it would seem, their own pigeon, / But it is quite novel at least.