Are there any benefits in training yourself to do basic tasks (brushing teeth and similar) or sports with your weaker hand. I've heard that some athletes tend to do that, so I guess there is a reason behind it.
I can't see how it could hurt? Not only will it be good for the muscles in that hand, it is also good for your brain. Using your right hand activates the left side of your brain, and uses the left hand activates the right side. Going further, any time you use your right hand to cross over to the left side of your body ("crossing the midline") helps the right and left sides of the brain work together.
It would build the muscles so that they are more even, so you don't have to rely so much on one hand. It would also be helpful to be able to use your weaker hand if you happen to hurt your dominant hand and can't use it for a while!
Yes, it's the best thing to do, using your weaker hand. You know, it balances your muscles and apart from that, if you keep using the stronger hand and ignore the weaker one, the one you perceived to be strong will soon start to become weak. So why can't we try using both?