Green’s Dictionary of Slang

a £5 note.

1861		(con. 1840s–50s) H. Mayhew London Labour and London Poor III 386/2: The notes were all finnies, (5l. notes).
1901		Argus (Melbourne) 20 Sept. 6/4: Bank notes are generally referred to as soft ’uns or flimsies, a five pound note being a finny, and a pound note a James.