I think the most appropriate solution to this is Cookies. If a user checks the remember checkbox then you must store a cookie with the name and password of the user. When a request from the same user comes the browser will also send cookies along with the request. So on the home page or login page you must check whether a username and password is attached with the request in form of cookies. If the username and password cookie exists, then you must log him/her directly in.
For security you must keep encrypted cookie on the client