When I was a little kid I wanted to play guitar, but there weren't many little-kid-size guitars in those days. But there were ukes - so, that's what I got. I didn't stick with it for all that long, and always regretted it. Seeing Janet Klein perform made me decide to pick it up again, because it looked like so much fun - and it is!
As for iconic - I get asked if I can play "Tonight You Belong to Me" a lot (or, more accurately, "that Steve Martin song from The Jerk" is usually how it's phrased) so that must stand out as *the* ukulele song to my generation.