Word for student's notebook

I agree with @JLG in his comment - we (US family with kids in school) call this a planner or student planner.

Composition book, perhaps? (The link is to a product search at Office Depot, so you can verify whether it's what you meant.)

In my son's present school, they have a book they call an "agenda" where the student is supposed to write assignments and that sort of thing.

I have four kids, the youngest now in his second year of high school, and they've attended schools in three different districts in two states, and no school has ever asked me to sign anything like that, weekly or otherwise. When I was a kid my parents were never asked to sign anything like that as far as I recall. Maybe that's a British thing, or if it's done in the U.S., it's not universal. Whether my kids' schools were the exception or the rule I have no idea.

The word 'composition notebook' is generally used in the US to describe one of these notebooks:

'Planner' is more widely used to describe what you're saying.