Barkarby (ESKB) is currently open on overtime from a municipal decision to close it down. It is still open, but call them and check current status before you go. No landing fees and parking should be free or very cheap.
Bromma (ESSB) is the most central airport but will cost you a little more in both landing and parking fees. I believe landing fee is about 300SEK (about 30 euros), parking fees are here
http://www.brommaairport.se/upload/B...AI-01-2009.pdf (excluding 25% VAT), scroll down for prices in english)
One option for you might be to land at Skavsta (ESKN), they have Ryan Air flights to London from the airport. Skavsta is "ryan air distance" from Stockholm, which means a rather long ride on an airport coach. I have no idea what parking would cost - call them. I think landing fee is around 300SEK here too...
Roslagens Flygklubb (Norrtälje, ESSN) is suggested, but it is rather remote with not so excellent communications (taxiride to stockholm won't be cheap). IIRC the the runway is 650m and in a slope, so that could be limiting depending on ac type.