Although solutions based on Russian policy....
are probably more interesting, the most plausible and efficient route to a persistent Russian Alaska is simply to have no other power choose to purchase or conquer the territory. The onus is on the US or Canada to take it over.
Simplest PoD - Seward's assassin succeeds in killing him in 1865, and nobody else in the Johnson administration is willing to buy the territory.
Americans may become willing to buy when gold is discovered there, but that is double-edged, as it will make the Russians more inclined to keep the territory or ask for a higher price.
I don't know how likely a Canadian takeover is. From the map, it looks geographically overdetermined, but then again, why didn't the British even bother to try to take it over during the Crimean War?
Maybe in a later Anglo-Russian war (itself a divergence, as there were none in OTL) Russian Alaska would not be so lucky. But Alaska is not worth it for London to start a war over, considering the other, more important territories that a Russo-British war would bring in to play, like Afghanistan, China and Persia.
Likewise, Alaska might look like a territory that the Russians could be willing to hand over as part of a global entente with Britain (a la 1907). However, the problem remains, what did the British empire have at that time that it would have been willing to give up.....for Alaska?
This leads me to some questions about the original sale of Alaska from Russia to the US. Did the Russians ever offer to sell Alaska to the US before the American Civil War? If so, why not? And in the lead-up to the historic sale, was there much haggling.
...and one last thought, regardless of what the Russians do if they have continued sovereignty over Alaska into the twentieth century, perhaps Mormons who dislike some of the compromises made to permit Utah statehood in the 1890s might like to move there.