"Bob's yer uncle" is usually explained as a slag on Robert Cecil, Lord Salisbury, who appointed his nephew, Arthur Balfour, to the prestigious post of Chief Secretary for Ireland in 1887. It is also said to originate in the expression "all is bob" (i.e., excellent), found in Francis Grose's 1810 Dictionary of the Vulgar Tongue.

I think the appointment of Arthur Balfour gave new life and form to the older expression.