In Wikipedia I read the Germans even occupied the place when they conquered Norway. So it's probably not the safest refuge for Jews.
A natural development of some coal and mining business slowly expanding also wouldn't be fast enough imo to help more than a few thousand people.
A better possibility might be a very rich Person foreseeing trouble in continental Europe and deliberately starting a predominantly jewish settlement there in the 1920s - even including souvereignity. Norway being the only country granting such souvereignity to a few of the Svalbard isles for a hefty price, a little exodus of a few thousand people in the beginning starts.
The place is bigger than Switzerland and should therefore be able to house a lot of people. But what are they supposed to do with a bunch of snow covered rocks in freezing climate and polar days and nights?
Trading is difficult as the harbors are probably frozen during parts of winter. Fishing is pretty difficult for non-professionals because of all the fog, ice and other bad weather-conditions. The competing professionals already there will also probably be difficult to deal with. Mining will probably not draw too many jews to this place, as at least in Germany Jews were pretty much excluded from such work and therefore rarely trained for it.
To feed a lot of people even under difficult economic and weather conditions, it would probably be necessary to create wintergardens in masses. I imagine some kind of cheap mass produced modules with living, transportation and working areas on the ground floor and a wintergarden on the top floor, covering both flat lands and southern hill and mountain sides. Inbetween enough space to let all the snow fall to and remove it if necessary.
While this would create all the jobs needed to draw more and more people to those Isles, it would also be very inefficient, especially at the beginning, so that standard of living would stay rather low. It might still be enough to draw a few hundred thousand people to the isles, considering economic problems and suppression of minorities in Europe.
But when in the 30s, conditions get even worse for the Jews and other minorities of Germany and some other countries, many more might go to this place. It might be offered as temporary quarter until they find a less hostile environment, to attract even more people. The ones who don't have enough money to pay for their journey to the isles, the stay there, and the journey to the next place have to work for a certain time to pay for it. The others will also be given work opportunities, so that more people can be given shelter. So the number of people there rises to a few million. At the end of the 30s, when the war breaks out, Jews and other prosecuted minorities from all over Europe come to the place, while defence is being upped.
With Svalbard not producing much more than it needs herself, extermination of the Jews not official policy of Germany yet, and not very many riches found there, the Germans might not even try to occupy the place after they took Norway. Instead, the Jews might make all the Svalbard isles their territory by joining Great Britain in the war against the axis. The British station some of their troops, ships and airplanes there, while the Svalbards give the British some much needed scientists, engineers, recruits, and so on. While the Germans are conquering France, the size of the Svalbard army is increased significantly, the English helping them in their training.
The Germans will therefore not be able to hide their ships in the area between Svalbard and Norway, which strengthenes allied positions there significantly and reduces efforts to find and eliminate German war ships.
Svalbard in the meantime builds up an army of small submarines and torpedo boats which help liberate the North Sea and later even the shipping lanes in the Baltics, so that the Germans have difficulties getting iron ore.
After the Germans attack Russia, Svalbard feels strong enough to attempt the liberation of Norway. The Germans don't have a possibility to put too many troops there, so resistance is within expectable limits. Svalbard lands in the north and fights her way towards the south, with some more surprise landings to encircle German resistance. Svalbard also advances towards Finland and either forces her to get a separate peace agreement with Russia or fight on two frontiers. While I do expect Finland to be too strong to be conquered by even a few million strong Svalbard, I expect it to be possible to keep Finland from attacking Norway successfully. Should Finland break the neutrality of Sweden to break the deadlock, Sweden should usually join on the allied side, which would also be fine. Once the Problem with Finland is solved, a faster route to help Russia is opened, so that Germany will feel even more pressure on her eastern frontier.
Once the US joins, Svalbard has even more material available. Submarines with mines, torpedos, cannons and later even rocket launchers regularly attack the coasts of Germany and her occupied territories. Planes make sure no German operations in the north stay undiscovered. The D-Day landings include a few thousand Svalbard troops. When the Allies reach Belgium, Svalbard and Norwegian troops liberate Copenhagen and other Baltic isles under German occupation. From there, bombing operations on Germany can be done much more effectively and frequently than from Britain, after air superiority is achieved. Due to all this and more, the war ends half a year early.
Interesting PoD for this ATL: Svalbard doesn't expect an attack on Russia and attempts the liberation of Norway too early, so that Barbarossa is actually cancelled or at least postponed once more. Germany is much better able to solidify her gains in Europe and therefore much more a problem for Britain and Svalbard. In that position, Germany might feel even stronger than in OTL though, and declare war on the US pretty directly after the Japanese attack Pearl Harbour. Maybe even earlier. That would even out odds again. May also be that Russia attacks some time in 1943 or so if they think they can get away with it.
