This is the second example of funny things which I used to hear said as I was growing up. In each case, I have trawled the internet together with a few books in my possession and compared the findings to come up with what I regard as the most plausible origin. I will always quote my source when there is no doubt but usually, as in this case, there are several options.
My poor father was once asked by my mother if he would build a brick surround fireplace in our living room. They were all the rage in the early fifties. My uncle had built one in his Solihull home but as with most of his DIY efforts it came to grief. It looked good enough when he'd finished it. In fact it was so impressive a structure that my Mum nicknamed it 'Warwick Castle'. Sadly for my uncle, it fell through the floorboards a few weeks after completion so he installed underfloor heating instead - but that's another story.
My Dad pointed out that the fireplace would be too difficult given my uncle's disastrous attempt but my mother would have none of it. "What is so difficult?' she asked. "Get some bricks, slap on some cement and Bob's your uncle". Except that Dad's uncle was Joe.
We often used this phrase - it meant 'easy peasy', 'job done', 'no problem' etc. etc. It seems that 'Bob's your uncle' was first noticed as a phrase in common usage during the early 1930's. This means that those suggested origins dating back to the 1800's are thrown into some doubt.
On balance then I will go with the widely reported theories which point to a music hall song written in 1931 by one John P. Long and sung by the famous Florrie Forde. It was called 'Follow your Uncle Bob' and included the lines
"Bob's your uncle
Follow your Uncle Bob
He knows what to do
He'll look after you"
Etymology? What can be so difficult about it? Just find the source and Bob's your uncle.