Green’s Dictionary of Slang

1. (bingo) the number one.

1917		A.G. Empey Over the Top 148: The caller-out has many nicknames for the numbers such as ‘Kelly’s Eye’ for one, ‘Leg’s Eleven’ for eleven.
1925		(con. WWI) Fraser & Gibbons Soldier and Sailor Words 122: 1 was ‘Kelly’s Eye’.
1930		(con. 1914–18) Brophy & Partridge Songs and Sl. of the British Soldier.
1936		(con. 1900s) F. Richards Old Soldier Sahib (1965) 70: No. 1 – Little Jimmy, or Kelly’s Eye.
1943		M. Harrison Reported Safe Arrival 83: The two Air Force corporals who ran the Housey-Housey game [...] burst into their esoteric chanting of: ‘Kelley’s Eye ... Shiny Ten.’.
1962		K. Waterhouse There is a Happy Land (1964) 87: Clickety-click, sixty-six [...] Kelly’s eye, number one.
1972		(con. 1940s) N. Conway Bloods 115: The caller had sung out his litany of Legs Eleven, Kelly’s Eye, Knock on the Door, Clickety-Clicks.

2. used adj., the best, first-rate.

1927

(con. WW1) P. MacDonald Patrol 139: ‘That first one [i.e. a prostitute] she was Kelly’s Eye’.