Totally depends on the listener.
My husband and I share the same taste, so that's easy.
For my mother I chose A New Day and a couple of "mother friendly" Oulu songs (Silver Bride, Sampo, Silent Waters, From The Heaven Of My Heart, Sky Is Mine and House Of Sleep). In my opinon live songs are always the best way to introduce a band to someone (if the band is good live that is (and Amorphis
is, and that's an understatement, too), if you want to introduce someone to The Cranberries you might want to skip the live performances...), because you can feel the energy and the enthusiasm of the band.
The first song I played for my dad was The Four Wise Ones, but mostly because we are in the video, otherwise it's not a very "dad friendly" song.

Songs I specifically played for him because I thought he'd appreciate them were Death Of A King an Sky Is Mine (Oulu).
If I had to pick three songs for a stranger, I don't know... There's so much awesomeness to choose from. Maybe Nightbird's Song, Death Of A King and From The Heaven Of My Heart?