My favorite hiking notebook is the Rite in the Rain 371FX-M. It weighs 20 grams, fits in my shirt pocket, and works fine when soaked with rain (or sweat).
Here are a couple of them that have been on a few treks.
I keep pretty basic notes: times, weather, campsites, and so on. If you write a lot, you might want more than 24 pages and maybe bigger pages. I use an official Rite in the Rain pen, but the paper is fine with most ballpoints. Pencils work, too. Sharpie pens smear, for some reason.
Pen and paper is one of the essentials, part of a first aid kit. If there is a serious incident, you’ll need to keep vital signs and notes. Those notes will go with the patient when they are evac’ed. You may also want to post a note at a trail intersection for directions.
Philmont encourages each participant to keep a journal. I bought a stack of these notebooks and gave one to each crew member. My son’s trail journal is almost all quotes, odd or funny things that other people said.
Here is a 3-pack of the 371FX-M on Amazon.