Green’s Dictionary of Slang

1. to wager on.

1885

Bulletin (Sydney) 16 May 14/2: Just before the Shorts Handicap came off, Citizen A. saw Citizen B. […] watching a chance for investing a pound on a ‘soft thing.’ Tripping silently over, therefore, he asked his chum what he was going to stand on, to which the latter replied: ‘Well, I’m going straight for No. 8 [Kalo]; best thing I’ve seen for many a day.’.

2. to trust, to rely on.

1880		Bulletin (Sydney) 3 Apr. 3/4: A young lady of Darlinghurst says that she never stands upon trifles. No! but she stands upon small nines.
1905		Wodehouse ‘Kid Brady — How He Made His Debut’ in Captain Sept. 🌐 You may stand on me. On the wor-rud of a mimber of the city police.
1906		Sun. Times (Perth) 18 Feb. 4/7: I ’ad a nappointment with me girl. Jim coulder come too [...] But ’e stood on ’is dig.
1925		Wodehouse Carry on, Jeeves 98: I knew we could stand on Jeeves.
1939		V. Davis Gentlemen of the Broad Arrows 10: Yez can stand on, boss.
1959		F. Norman [bk title] Stand On Me.
1961		(con. 1920s) J. Sparks Burglar to the Nobility 47: You can stand on me that this is true [...] for I am a man of experience.