Here are the reasons I notice students quit and my advice to avoid the pitfalls:
Not making practicing piano a daily routine
This reason is related to the first reason. For a student to stay with piano when they're under 7, the parent needs to help the student build practicing piano into their routine. It's easy for a 7 year old to come home from school, watch TV, do some homework, play video games, and then go to sleep. If practicing piano daily isn't built into a routine at the very beginning, it's hard to become disciplined enough to practice. By the time the student, who practiced only once a week, has learned piano for 2 years, they might feel like they have made no progress, so they drop out.
Not having a great teacher.
Teaching is a talent just as much as the ability to play an instrument. Having a bad or mediocre teacher can drive the joy out of anything. Having a great teacher can be life changing. You need someone who not only can teach you skills but who also inspires you. You need someone who is going to challenge you, most importantly. The worst thing a teacher can do is only give you things you’re comfortable with or things below your potential.
Not recognizing that there will be ups and downs in enthusiasm, commitment and rate of learning. Keep at it over the long haul and don't quit during a down. Change it up when you get in a rut. Look for ways to re-spark your interest-play music in a group, perform, write, record, go hear live music or listen to music you love.
Not making realistic plans on how much practice will happen and then feeling guilty.
People then quit because they want to stop feeling guilty. It is much better to make a plan you can actually live with over the long haul rather than make a plan you can't stick to. Forgive yourself when you get off routine. Remember that learning music should be something that makes you feel good not be another thing to feel pressured about. YOU have to decide how you want music to fit into your life and then be ok with that. If you can't practice much, accept that your progress will be slower.
Not making concrete plans on when practice will happen
Attach practice to another already existing routine so there's not a decision to do it. Don't just wait to FEEL like practicing. Something will always get in the way.
Parents expecting children to be motivated and disciplined to go practice on their own. Discipline is learned not discovered.
Parents leaving the decision to take lessons or not up to kids.
Kids are not ready to make smart long term decisions yet. They are very much in the now. No matter how much they enjoy music, there will be a time when they don't want the work or the responsibility. Most likely, they will regret quitting as an adult. Parents play an essential role on a piano students' progress at this age. This means that the parent needs to give lots of encouragement and patience. This is the age when kids still, for the most part, listen to their parent.
Not doing music the student loves as soon as possible.
Skills and theory can be taught through music and music can be tailored to fit the level of the student. It shouldn't be a far off goal.
Not having clear goals on what the student wants to get out of it.
What kind of music do you want to do? One on one lessons should be customized. What skills to focus on might be different depending on what the student wants to be able to do. Sometimes a student doesn't know what they want and needs to be exposed to different things until something ignites the spark.
Students being overcommitted and not showing up regularly with materials.
I've seen plenty of students still make progress even if they don't really practice but not showing up regularly for lessons on top of that make it impossible to move forward. If there is too much time between using information, the brain will use that brain real estate for something else.
The novelty wears off
It's easy to be in love with the idea of something and then the reality of the work sets in. It's a good idea to move as fast as you can through the 'novelty' period so that by the time the newness has worn off, the student is past the first bump and actually able to play fun things.
Middle school and high school are a time when many students quit.
The added social and schoolwork pressures often push piano lessons out. Students also compare themselves more heavily to others who may play piano and feel embarrassment or not good enough. This is a shame because this is the point when they're usually capable of the most. If students can get a solid foundation before this time hits, piano practice can be a stress reliever instead of a stress-causer.
Student can't accept that struggling is part of learning.
Adults sometimes forget this is part of learning and can't bear to have their child feel uncomfortable. However, if they are not challenged at all, they don't improve, get bored and quit. Sometimes people want to believe they can learn the hard thing without doing the hard part.