Using the notes on the D string and soon after the A string you can teach most of the music in Essential Elements for Strings. The following songs are favorites of the student:

#17 Rolling along - notes on D string only.  Work on hand shape and tuning.
#22 Tunnels - work on curved fingers.  Before they start to play have students pluck A string while holding down fingers on D string.
#24 Lightly Row - another "tunnels" song 
#40 Mozart Melody - otherwise known as Twinkle.  Teaches student about ABA form and can be used to teach variations, including students making up their own variation.
#40 Twinkle Variation - this variation is called Pepperoni Pizza or Mississippi River, there are many other variations and you or the student can make up ones on your own.
#81 Can, Can - I use this to help students learn to play slow to fast.