The only possibility I see is for it to be a puppet, sponsored by someone - eg if Russia in the nineteenth century spreads itself over all of the North of Scandinavia, then comes a war with Britain and France (analogus to the Crimean War) and after this war as much as possible of Russia is broken off, then perhaps just perhaps you would get a Lapp state which is basically a British client. Of course, Britain would have to be very angry at Sweden to do this!
Grey Wolf
It would be pretty easy to think up somewhat different Napoleonic Wars that in the north lead to a lot bigger chunk of Swedish Lapland and even some of the Norwegian coast being annexed by the Russians by the 1820s. Seeing the strategic potential of the new territories on the Atlantic coast, the Russians split this Lapland-Finnmark into a new administrative entity, directly controlled by St. Petersburg while the rest of the conquered area becomes a Grand Duchy of Finland.
Then, let's allow several decades of peace so the Russian rule up north is solidified. Russian interest in the area is almost purely military, and apart from developing ports and roads, later railways, the Russians adopt a pretty hand-offish attitude to it - like to the Finnish Grand Duchy IOTL. This Lapland-Finnmark becomes a distinct entity from Sweden, Norway, Russia and Finland, and after a while becomes considered a Sami homeland. Because of the low population, immigration is encouraged even from abroad - this is helped by a Lapland gold rush sometime in the 19th century. If Lapland-Finnmark is known abroad, it is consided the "European Alaska", a cold new frontier with natural beauty and wild natives.
After this there would be several possibilities how this Russian Lapland-Finnmark could become an independent state - ostensibly, because the low population and limited natural resources (until the late 19th century when the mineral deposits are uncovered) would mean that it would need a protector, whether that is a Scandinavian/Nordic power, Russia, Britain, Germany etc.