As someone who's ambidextrous, I'd highly encourage letting him go with what is natural.
I'm the only person who is a lefty in my family, but all I do left handed is write and eat. Everything else (throwing, gymnastics, brushing my hair/teeth) is predominant right. It's allowed me to do many things I'd never have been able to had I truly been dominant one way or another.
There have been times being left-handed has been challenging. Picking where to sit at restaurants, using desks in college......but, if you've ever noticed, most brilliant people are predominant left. Check which hand your physician writes with, your pharmacist, etc.
If you believe that hand dominance is truly tied to different parts of the brain, I gladly welcome the diversity of being multi-gifted. I wish one of my kids were dominant left, but our daughter is showing a proclivity to both left/right dominance, and I love it.