"Bob’s your uncle" is a way of saying "you’re all set" or "you’ve got it made." It’s a catch phrase dating back to 1887, when British Prime Minister Robert Cecil (a.k.a. Lord Salisbury) decided to appoint a certain Arthur Balfour to the prestigious and sensitive post of Chief Secretary for Ireland. Not lost on the British public was the fact that Lord Salisbury just happened to be better known to Arthur Balfour as "Uncle Bob." In the resulting furor over what was seen as an act of blatant nepotism, "Bob’s your uncle" became a popular sarcastic comment applied to any situation where the outcome was preordained by favoritism. As the scandal faded in public memory, the phrase lost its edge and became just a synonym for "no problem."

From: word-detective.com