Stalin's arch rival Trotsky was exiled, but why did Stalin wait until later before sending assassins to finish him off?
He was exiled to Turkey by way of Khazakstan and not Siberia - and it wasn't until the Show Trials until Stalin felt he had consolidated enough power to order Trotsky's death.
I'm not sure there's going to be a definitive source for an answer to this.
From a (sociopathic autocrat's) political standpoint, I do think it was good timing. If he'd done it earlier when the guy was still in the news and had lots of followers, he would have created more of a martyr.
Better to let things cool down for a while, perhaps spend a few years finding and purging any well-placed followers he had and letting the state propaganda system have some time to lessen any following he had among the people. Then, when the guy's bascially already a non-entity, kill him to show the world what you do to folks who go up against you.
In 1936 Trotsky was convicted by a Moscow court to death in absentia. After that the 4 years before his assassination was just a technical period of preparation for the perpetration of the sentence.