I've recently started learning to spin poi (it's not important; you may think about dancing) and found that my left hand requires a great deal more attention to follow the trajectory I have in mind than my right one. This is extremely pronounced before I form some muscle memory, and becomes somewhat less severe after. Obviously, the only way to get better is a lot of practice, and that's what I intend to do. However, while thinking about this problem I visited quite a few websites with the advice on how to improve the coordination of non-dominant hand. Some of them recommended writing by that hand. I doubt that developing fine motor skills will help me with the complex hand movements heavily involving elbow and shoulder. So I decided to ask for a reference, preferably to some scientific paper (this hypothesis looks easily testable, so such papers probably do exist). I was unable to find anything on this topic, but maybe you could give at least a personal story.
The carryover from one activity to another is generally better the more similar the activities are. In sports, this is know as the Specificity Principle which states:
So to get better at one thing, you should preferably do that exact thing.
That doesn't mean that you won't benefit from similar activities, though. Generally, the more similar the activity is, the greater the carryover. Do keep in mind though, that similar activities put similar strain on similar structures, so they can actually be harmful.
Now to answer your question, is writing with your weak hand similar to spin poi moves with your weak hand? As far as I understood spin poi, not really. Writing is slow and deliberate and you're holding the pen in your hand in a fixed position. This is significantly different to spin poi with its dynamic motions and frequent finger twirling, so at least mechanically there's practically no benefit I can think of.
Neurologically, though, both activities may activate similar parts of your brain, which could mean that there's at least a neurological carryover. Although I doubt it, as spin poi is about coordination of both hands (even arms) at once, while writing is about coordination of the muscles in one hand. Anyway, I'm not near smart enough to answer that question, so I guess the jury's still out on that one.