This article is from the alt.usage.english FAQ, by Mark Israel email@example.com with numerous contributions by others.
This British phrase means "all will be well" or "simple as that":
"You go and ask for the job -- and he remembers your name -- and
Bob's your uncle." It dates from circa 1890.
P. Brendon, in "Eminent Edwardians", 1979, suggests an origin:
"When, in 1887, Balfour was unexpectedly promoted to the vital front
line post of Chief Secretary for Ireland by his uncle Robert, Lord
Salisbury (a stroke of nepotism that inspired the catch-phrase
'Bob's your uncle'), ..."
Or it may have been prompted by the cant phrase "All is bob" =
"all is safe."
(Info from Eric Partridge's "Dictionary of Catch Phrases", 2nd
edition, revised by Paul Beale, Routledge, 1985, ISBN