Ok, so this is more of a treat day than entertainment on a budget but I took my 5 year old to see Tréð today at Tjarnarbio theatre. My Icelandic is still a bit basic so don´t test me on the plot, but we really loved it.
General ticket price for family shows in Tjarnarbíó is 3.100kr. If you sign up for their mailing list you get a discount on shows and can keep on top of the shows they have for family / children. There are lots of options to get discounted tickets. (check their website for this) I got 20% with the Reykjavík Cultural Card but you can get up to 50% discount using some of their loyalty cards!
It's a lovely space to meet friends; with snug areas for sitting with coffee and snacks. There is a also a cafe and a bar.
Address: Tjarnargata 12, 101 Reykjavík