Green’s Dictionary of Slang

a flogging with a strap.

1833		Chelmsford Chron. (Essex) 12 Apr. 4/4: The First of April. Joseph Lamb [...] was charged with assaulting a boy named Cadman. — The boy was sent to defendant’s shop in the morning of this day of wit, for a straight-hook, but by some mistake he was supplied with strap-oil.
1847–81		Halliwell Dict. Archaic and Provincial Words II 816/1: strap-oil. A severe beating. It is a common joke on April 1st to send a lad for a pennyworth of strap-oil, which is generally administered to his own person.
1857		Worcester Jrnl 10 Jan. 7/2: Prosecutor admitted having struck the prisoner with a stirrup leather [...] The chairman: Better known as ‘strap-oil’.
1874		Fife Herald (Scot.) 15 Jan. 4/2: Come, old Macaroni [...] haven’t you given that younker enough of that strap-oil?
1895		Coburg Leader (Vic.) 20 July 1/6: Since N.H. has been feeding on pigeon’s milk and strap oil he is getting quite fat.
1890–1904		Farmer & Henley Sl. and Its Analogues V 92/1: oil of [...] strappem [...] a beating.
1896		Manchester Courier 13 Aug. 5/5: The old Lancashire custom of sending a young apprentice to the shoemaker’s for ‘strap-oil’ [etc.].
1900		Bulletin (Sydney) 8 Dec. 22/4: ‘Now, Miss, I’ll fix her up for the night,’ went on Mrs. Davis; ‘then I’ll give Rex his ‘Cherub’s Balm’ and turn-in myself. The lodge doctor was that disgusting, saying the child must have gorged himself on the sly with green gooseberries, and I’d better give him ‘strap-oil’ and mustard-water. Why Davis joined the lodge to have one insulted, I don’t know.’.
1907		C.E. Mulford Bar-20 xx: Thereupon they treated their mounts to liberal doses of strap-oil and covered the ground with great speed.
1939		P.G. Brewster ‘Folk “Sayings” From Indiana’ in AS XIV:4 267: An obstreperous youngster is likely to be dosed with ‘peach tree tea,’ ‘hickory tea,’ or ‘strap oil,’ or by some other means forced to ‘walk the chalk’ and to ‘toe the mark’.
1940		M. Marples Public School Slang 132: A type of joke once popular in many forms — e.g. a boy might be sent for a penn’orth of nothing with nobs on, or some strap-oil (=a beating) .