I think the question is completely wrong, it's not how they should remember their passwords. It's why do they have several usernames and passwords in the first place?
First the resources that are school controlled should of course be behind one username/password pair, preferably SSO for the web parts (e.g. a CAS variant is quite simple).
For external resources, is there a real reason they really need to log in? E.g. can IP based access control or something work for some cases. I understand you don't control everything, but as users(/customers) one can at least complain, and try to push it in the right direction. If there is a reson to log in, do they support something like Shibboleth/SAML or OpenID for login federation? If so, that should be used. It's not trivial, but making the lives of the students hard for something stupid like that is even worse
I think that for an elementary school student, if the amount of username/password pairs they need is over 1, there's something wrong somewhere.