Light and cheap? Can you keep it simple with a simple notebook and quart/gallon sized ziplock? Chances are, if it's going to rain anyway, you're already packing waterproofing for the sensitive contents in your pack?
Depending on how much writing you plan on doing, you might consider putting together your own. A small variety of shipping "envelopes" (bags, really) are made from similar material--as the Rite-in-rain--and the plain-white inside can be used. IIRC, in the past, UPS (or was it USPS?) would give you lots of these bags/envelopes [free] if requested; I'll let you decide on the ethics of that if they are willing.
Either way, unless you're using a pencil, you may want to experiment with different ink pens. I've found that some inks stay put and some don't, regardless of the paper type.