Before "retiring" from teaching in order to run Henderson Academy of Music full-time, I taught A LOT of piano lessons. I taught piano for about 8 years (I started around age 16 when I was in high school). and have taught hundreds of students- both in LA and in Seattle- and thousands of lessons.
I think the most important thing I've learned as a teacher is this:
Music lessons must be enjoyable if you want the student to continue taking lessons.
And I think that's just what most parents want...for their kids to continue taking lessons and be consistent. After all- if you stick with music lessons for several years, you'll start to get pretty good!
Anyway- I really think that it's important to find a teacher who will make lessons FUN. This goes for kids as well as adults. Most adults want to learn piano just for the fun of it. Now, of course, it isn't all fun. Practice can be tedious and/or frustrating at times, but I think it's worth it.
And for kids, the benefits of music lessons are undeniable (I'll probably write a post about this at some point). Music lessons teach kids discipline, the value of hard work, performance skills, and music lessons supposedly increase math test scores (though I think this is largely overblown).
So- if lessons aren't fun, the student won't continue lessons- and won't get the benefits. Now...how to make lessons fun?
I think the first factor is the teacher's personatliy. Especially for kids. Kids need someone who will keep them engaged and excited about music. I think personality is almost as important as musical ability when it comes to teaching.
The second factor is the repoertoire- the pieces the student learns. If a student is playing songs he/she likes, he/she is more likely to practice, more likely to improve, and more likely to stick with lessons!
So, parents...make sure you find a teacher for your child who has a great personality and will teach your child songs he/she likes to play :)