Some ways learning music can be good for your child:
--It can make your child a future income! Although I started out playing for church for free back in my First Baptist Church, Monroe, Louisiana hometown, it eventually developed into a paying gig when I became the organist of Lakeshore Baptist Church, also in Monroe, when I was 16 years old. So literally half of my life has been spent playing for churches, and I estimate I have made more than $100,000 over my lifetime from playing for worship services, weddings, funerals, etc. Seriously, that's a true number. And yes, I enjoy playing so much I WOULD do it for free (which is always the question you should ask whenever you wonder if you love your job).
There are so many reasons beyond these listed... I literally wrote my college Honors Program 40-page thesis on why music lessons are good for a child's brain development. That was a decade ago, so I don't remember everything I wrote! Maybe my brain's memory capacity hasn't been developed by music lessons as much as it could be, LOL!
But there are few people who would argue that music lessons are not beneficial-- so please consider supporting your child with lessons if he or she shows an interest in music!