Consider a song with a lot of variety. Your song should have changing tempos, a large range and different styles.
Ensure that your song appeals to both modern pop and older musical sensibilities. You want to show that you can sing modern songs and classic songs. If you're choosing two songs, then you should pick one from each category.
Familiarize yourself with the roles of the musical, and be sure that your song coincides with the role that you want. For example, if you're trying out and desire the part of Liesel in "The Sound of Music," then you may want to pick a melodious and cute song to sing. You wouldn't want to pick a raucous and angry tune, or the casting director might think that you don't fit the part.
Sing something that you can easily memorize. Don't pick something that has complicated lyrics or melodies that you cannot remember, as this will suggest to the judges that you have difficulty memorizing things.
Pick a song that allows you to act dramatically and dance. You don't want to bore the casting director by standing still on stage. Picking a song with different parts or emotions allows you to be animated on stage and move around.