Green’s Dictionary of Slang

tibby drop n.

In phrases

drop on someone’s tibby (v.)

to take someone unawares.

1873

Sl. Dict. 321: To drop on one’s tibby is to frighten or startle any one, to take one unawares.

on one’s tibby drop

unawares.

c.1864		A. Stephens ‘The Chickaleary Cove’ 🎵 I’m a Chickaleary bloke, with my vun-two-three — / Vitechapel was the village I was born in; / To catch me on the hop / Or on my tibby drop / You must vake up very early in the mornin’!
1875		Appleton’s Journal (N.Y.) 1 May 562: Goodbye johnny, before I leave you, / One more kiss before I go, / For to catch me on the top, / Or on my tibby drop.