Nothing leads to a student quitting piano lessons faster than when the lessons and practice session become chores. Create an inviting learning atmosphere each time a student comes in for a lesson. Having a successful lesson doesn't mean that she plays through a song perfectly. Instead of focusing on technique, try enjoying the piece of music with her for what it is. Allow her to share what she is thinking about and what emotions it invokes her to feel as she plays the song.
Words of encouragement go a long way for someone who may be struggling with self-esteem. If a student exhibits obvious musicality but doesn't seem to be progressing with his lessons, perhaps he needs an extra boost of confidence. Build him up by pointing out all the things he is doing right during his lessons. You need to be honest about correcting technical mistakes and posture. But you can work those in gently, then follow with additional compliments.
Occasionally, you may have students who give great performances during recitals but do not seem to be as engaged during lessons. These students enjoy being the center of attention so they may find individual lessons boring. Allow such students to shine by setting up mini-concerts each month for your students. Assemble different groups from month to month, so students are performing for a different audience each time.
Recognizing progress continuously helps provide that extra stream in motivating students to continue lessons. When a student knows she is doing well, she naturally becomes excited to tackle her next task at hand. A concrete way to track her progress is to keep a running record for the number of individual pieces she can play through entirely without stopping in the middle or starting over. As you work with your student on each piece, she will start to learn what your criteria are to earn her a passing song, and she will be more likely to adhere to those criteria to ensure she continues to pass additional songs during subsequent lessons.
Letting students choose one or two songs to work on each month during lessons allows them to express their personalities. Music is to be enjoyed, no matter what genre you prefer. When you open the door for students to choose their favorite songs, you propel their excitement for learning and enjoying music. As long as they can provide you with sheet music that has been written for piano, help them master those pieces. They will be excited to master some of their favorite songs.