Many people wonder exactly what kind of person their child will grow into.  And, along those lines, many people have heard about the many academic and cultural benefits private music lessons can provide. (For particulars of these benefits, please visit this page at the Society for Neuroscience web page.)  I occasionally meet parents who wonder why they should go through the frustration of "laying down the law" when their children reach a stage where they feel like stopping music lessons. 

So, what are the benefits of sticking with it?

Music lessons benefit the whole person. These benefits may be categorized as mental, physical, emotional, and spiritual.
Most of us naturally associate music-making with mental activity. Reading, analysis, and memorization all fall into this category. However, singing or playing an instrument is also a physical activity. Coordination, posture, timing, and poise are all benefits that result from long-term music study.
For students who participate in recitals, confidence is a key emotional benefit. The courage it takes a normally quiet introvert to publicly pound a nine-foot piano is considerable. For the rest of the year, studying music is a journey that offers many obstacles to be overcome. Patience is a virtue we normally associate with practicing, but what about the consolation it offers after a day filled with troubles? The routine of daily practice is objectionable to some, but it offers stability to adolescents when everything else seems to be in a state of perpetual flux.
The spiritual benefits of learning music are, literally, spiritual values learned from the music itself. Beyond learning an instrument, then, learning music means learning how music behaves, and then applying that knowledge to life. Most music has a beginning, middle, and end—so does life. Many pieces travel through darkness to light, from difficulty to triumph, or from conflict to resolution—these are all models that are imitated in life, ideally.
Finally, there are the benefits of communicating the values learned through performance. This could be a studio recital, but more likely the communication takes place at home, for friends and relatives. Whether in the living room or Carnegie Hall, music-making means taking the support of parents, friends, and teachers, and giving back to the community by playing life-giving music for them.

When a music student quits, everyone loses.

If bigger brain  parts mean a bigger intellect, musicians may  have a leg up on others.  Brain imaging research shows that several  brain areas are  larger in adult  musicians than in non-musicians.