this should get you started: http://www.quirksmode.org/js/cookies.html
the following examples use the functions declared in the above link.
creating a cookie:
// when the div is clicked
createCookie('hideSuggestionBox', 'true', 1);
reading a cookie:
// when deciding whether to show or hide the div (probably on document ready)
if (readCookie('hideSuggestionBox') === 'true') {
// do not show the box, the cookie is there
} else {
// the cookie was not found or is expired, show the box
}