Firefox: Remember new passwords without prompting

Try:

Edit → Preferences → Security → Passwords/Exceptions

and remove the site from the list in the dialog box. Also make sure that "Remember passwords for sites" is marked.

OK so here is another answer I think is more convenient that the Javascript one.

First go to your Firefox components folder, by default in my case it's "C:\Program Files\Mozilla Firefox\components"

Now open the file nsLoginManager.js

Go to the function at around line 804 (varies with version of Firefox), that reads:

/*
 * _isAutoCompleteDisabled
 *
 * Returns true if the page requests autocomplete be disabled for the
 * specified form input.
 */
_isAutocompleteDisabled :  function (element) {
    if (element && element.hasAttribute("autocomplete") &&
        element.getAttribute("autocomplete").toLowerCase() == "off")
        return true;

    return false;
},

Now comment out the first 3 lines, so it ends up like this:

/*
 * _isAutoCompleteDisabled
 *
 * Returns true if the page requests autocomplete be disabled for the
 * specified form input.
 */
_isAutocompleteDisabled :  function (element) {
    /*if (element && element.hasAttribute("autocomplete") &&
        element.getAttribute("autocomplete").toLowerCase() == "off")
        return true;*/

    return false;
},

Save it, restart Firefox and you should be good to go...