Keepsake Kitsch: Buy every souvenir in the gift shop. This will cost you a total of 38,620 kronur (unless you use the eggs to trade for stuff, which you can't do for the book, the batteries, or the sword). Since you can only have 9,999 kronur at one time, you'll need to make money, buy stuff, then go make more money to buy more stuff.
Spelunker: Enter all the rooms in the ice cave. In addition to the places where you need to go to find specific objects, there are a couple of dead ends. As you go through, every time you stop, scan around for arrows pointing in surprising directions.
Spoiler Free: Don't read any in-game hint all the way to the end.
Trivia Tamer: Get the trivia question at the end of the game right.