The most standard song that is known to sound amazing when played in the ukulele is somewhere over the rainbow by Israel Kamakawiwo'ole. There is something about how this song is played on the ukulele that makes it amazing.
More and more people are starting to learn more about playing the ukulele so they may not immediately realize anymore if a song is exclusively played in a ukulele or it is something that used to be played in other string instruments. There are some books with a list of songs that are better played with a ukulele. You can familiarize yourself with these songs as well so that you will realize why they are better played with the ukulele.