Nobody is really sure where this expression came from, but it was first heard publicly in the 1880s.
Prime Minister Lord Salisbury, Robert Arthur Gascoyne-Cecil, noticed his impressive young nephew making his mark on the Conservative government. Arthur James Balfour showed great promise and Salisbury appointed him to a range of positions including Chief Secretary to Ireland, Secretary for Scotland, First Lord of the Treasury and leader of the House of Commons.
Balfour then replaced his uncle Robert as Prime Minister in 1902.
It seemed that Balfour was capable of almost anything and gained all sorts of positions. The ‘word on the street’ was that “nothing is a problem” when “Bob’s your uncle”.
Travel Trales 
May 1, 2013 at 8:28 am
Interesting to know that! I have a slightly different take: when men have a sex change, they say, A nip and a tuck, and Bob’s your aunt!