Nobody's listed anything yet.
My list is eclectic, but it's mine. Listed in no particular order.
"All My Loving" - I'd wanted to be a Beatle since, well, seeing A Hard Day's Night when I was very young. Being able to play some of my favorite Beatles songs was a primary motivation for learning to play a stringed instrument.
"Across The Universe" - Another one of my favorite Beatles songs.
IZ's "Over The Rainbow/Wonderful World" - A trite selection perhaps, but when I began playing four years ago, this song seemed to be everywhere and it was sort of the "popular music" face of the ukulele. I also thought that if people knew I played the ukulele, it'd be a song that would get requested a lot.
"Ke Aloha" - My search for group ukulele lessons led me to a Hawaiian music class, where I learned to play two of the most basic Hawaiian strums by learning to play "Puamana" and "Waikaloa." "Ke Aloha" is one of my favorite Hawaiian songs.
"Tip Toe Through The Tulips" - I know, it plays into the whole "Tiny Tim ruined the ukulele in the 60s" thing. Because so many people associate it with the ukulele, though, I thought it would be a good song to learn.
"Come Monday" - I remember how much I loved this song when it first came out when I was in college, although I don't remember why. It's one of the songs I promised myself I'd learn if I ever played a stringed instrument. It's not a "must-play-on-the-ukulele" song per se, and, in fact, I prefer playing it on a tenor guitar, but I learned it on the ukulele and being able to play it at all has brought me a lot of joy.
"Donald and Lydia" and "Hello In There" - My girlfriend in college introduced me to the music of John Prine and these were two of her favorites. The original lyrics of "Donald and Lydia" are a tad un=PC (I've changed the lyrics to reflect that Lydia is shy instead of fat), but there's a beautiful simplicity to these songs that has given me a new appreciation for Prine as a songwriter, these two songs have great sentimental value, and I've been using "Hello In There" to practice fingerpicking techniques.
"Adios Ke Aloha" - Learning to play the ukulele has opened up a world of Hawaiian culture to me. This song, in which the song's protagonist realizes that he has lost a love he may never have had to begin with, has become a recent favorite, probably due to extenuating circumstances.
"Wahini Ilikea," "Peaceful Easy Feeling" and "Brown-Eyed Girl" - I've become a big fan of Rev. Dennis Kamakahi, a prolific Hawaiian songwriter and slack key guitarist who writes beautiful songs. I sometimes attend a monthly Kanikapila in Alameda, California (near Oakland). Whenever we play "Wahini Ilikea," we invariably segue into both "Peaceful Easy Feeling" and "Brown-Eyed Girl," which have very similar chord structures. Three great songs at once!
I realize I may not have answered the question the OP asked, but these are among the songs that are most important to me as an ukulele player.