Nokia was certainly part of it. Even if they failed with their N-Gage gaming phones, some of the current successful companies like Rovio started around that time. Nokia spent money and convinced some state agencies to do the same. Nokia-effect also bloated CS&E programs in higher education. No direct links can be drawn, but these opportunities to learn must have contributed something towards the current state of mobile gaming industry.

Very active demoscene-culture in the 90's was incubator for the first Finnish gaming companies. For example Remedy (Max Payne, Alan Wake), Futuremark (3DMark), Bugbear (FlatOut, Ridge Racer Unbounded) and Housemarque (Super Stardust, Dead Nation). http://en.wikipedia.org/wiki/Demoscene

Then there was National Broadcasting Agency (YLE), which in 1985 started a radio program called Silikoni. They even sent BASIC code nationwide via the air waves, which you could tape and then use in your computer. Readers could also send their own programs, which could appear in the program. Over 120 000 listeners nationwide were tuned in 1986. Here's one program (not sure if it is geoblocked, 4.15 should be where the code begins): http://yle.fi/elavaarkisto/artikkelit/silikoni_lahetti_radio... So I guess when people grow up with these things around...

I suspect that it's also somehow related to Finland being very good with open source projects. For example 1/2 of LAMP stack originating from Finland.

Cold dark winters with nothing else to do than play or make computer games.

Superior education system that focuses on creativity and completely devoid of competition. I believe they are number one in world ranking in that regard. If you want to look at their methods, there's a book called Finish Lessons.

Nokia.

Plus some very enlightened Finnish policies designed to answer the question, "Where is our next Nokia going to come from?" Policies that go well beyond offering tax breaks in some government-designated "technology and biosciences hub".