Urinate. Jimmy Riddle = piddle.
Cockney rhyming slang. The term is mentioned in Partridge's A Dictionary of Slang and Unconventional English, 1937, where it is listed as being 'late 19th century'. Many rhyming slang terms that refer to names derive from real people - the celebrities of their day. A recent (1980s) example is a 'desmond' (a second-class degree - a 2:2, derived from Desmond Tutu). From the late 19th century we have 'on your tod', which refers to the American jockey 'Tod' (James) Sloan.
It's not likely that Jimmy Riddle was a real person. If he was then it would be surprising that he were well enough known to have had a phrase coined in his honour only for any records of his existence to be now missing. Several 'Jimmy' phrases were coined in the 19th century; for example,
- Jimmy Grant (immigrant/emigrant)
- Jimmy Ducks - a sailor in charge of livestock onboard ship. This is a similar coinage to that of a character working in the prison kitchens in the BBC comedy Porridge - 'Luke Warm'.
- Jimmy Woods - someone who drinks alone.
- Jimmy O'Goblin - a sovereign.
These use the name Jimmy as a generic man's name - much as Londoners and others now use John, and that's probably how Jimmy Riddle derived too.