I wouldn't worry about the pen. The real issue for someone who uses a pen outdoors in all weather are the ink system and the paper.
There are waterproof 'papers'. Inks are another matter. Conventional ballpoint ink is water resistant, but the principle of the ball point relies on the friction between the ball and the paper. That could be a problem if things are wet.
There are waterproof inks for fountain pens - but AFAIK, these are only waterproof after they have dried.
Roller ball pens need less friction, and probably would work better that ball points. But the ink in them may not be waterproof. I think the best option would be a rollerball that uses a waterproof (or at least water resistant) ink.