The great thing about the ukulele these days is that a lot of popular songs that we're originally played on this great instrument are being covered on it--you can find uke videos all over the internet for most popular songs today. But for those interested, here is a list of famous ukulele songs that are well known for being played on a uke, and ones that a lot of people will probably be asking you to play once they find out you have the instrument.
Tonight You Belong To Me is probably going to be on every list you find. This song was written by Billy Rose and Lee David in 1926, but that's not how most people know it. It became popular in the movie "The Jerk" where Steve Martin and Bernadette Peters sing along. Since that time there have of course been many covers.
Another older song that keeps getting covered today is Ukulele Lady, originally by Gus Kahn and Richard Whiting in 1925. You can find popular covers by Kermit the Frog, or Better Middler.
Somewhere Over the Rainbow is one of my favorite famous ukulele songs. It was originally done by Harold Arlen and Yip Harbury in 1939, but the version that really made is a uke song is by Israel Kamakawiwo'ole. If you don't think you're familiar with the uke version, you probably heard it in the movie 50 First Dates.
There is no uke played in Jason Mraz's song "I'm Yours", but people cover this song on the uke constantly, making this a popular uke tune. This is a great example of how a song that's more popular today is helping making the uke a more common instrument.
Hey, Soul Sister by Train is another more current popular option. This is one of the very popular, current famous ukulele songs that gets covered a great deal online and that your friends will probably ask you to play shortly after finding out you have the instrument, so it's worth learning now.