Lord Salisbury
Turns out that no one really knows where or even when the phrase came to be. One theory says that it derives from the phrase – all is bob. Once again meaning that all is well. The phrase all is bob was found in Captain Francis Gorse’s Dictionary of the Vulgar Tongue, which was printed in 1785.
The most popular theory is that it derives from an anti nepotism sentiment as a result of Lord Salisbury appointing his nephew Arthur Balfour to many jobs he was unqualified for. Eventually Balfour was appointed chief secretary of Ireland. During the Victorian era this was an extremely sensitive position which he was unqualified and unsuited to. One of the great British pastimes is to make fun of those who have proven themselves to be out of their depth. So this irreverent poke was invented to say that having Bob (Robert Cecil, Lord Salisbury) as your uncle was a guarantee of success.
Another theory says that it is actually Scottish in origin. There was a 1924 musical called Bob’s Your Uncle according to an ad in the Scottish newspaper, The Anges Evening Telegraph.