Learn to play the guitar or write with your non-dominant hand and you will learn to play the guitar or write with your non-dominant hand. Actually, this is where the visible results end. Your question, as I understand it, is based on an outdated version of the concept of interhemispheric asymmetry (left-hemispheric logics and right-hemispheric dreamers - no matter how beautiful they sound, they do not correlate with reality). The benefits for the brain of this activity are not so dubious, but not experimentally proven. It is much more useful in this sense to study another language - there your whole brain will work at once, and the risk of developing Alzheimer's disease will decrease.
The main thing is not to take an example from one of the four that is on the avatar of the author of the previous answer: take a guitar and learn how to play it, but the neck should be on your left. And if the neck is on the right, then you will be like McCartney and you will not develop your left hand.
The point is that you will develop your second brain hemisphere, well, and become more developed in a sense. And about how to develop: Do all the usual things not with your right, but with your left hand, i.e., for example, try to eat with your left hand, brush your teeth, brush your hair, write in the end. At first it will not be familiar and it will be bad at it, but if you want to learn, then you can handle it all!