Install the following add-on. If it doesn't help, post a link to the login page.

• https://addons.mozilla.org/firefox/addon/rememberpass/

That website may be using autocomplete=off to prevent Firefox from saving form data like name and password.

You can remove autocomplete=off with a bookmarklet to make Firefox store form data like names and passwords.

• http://kb.mozillazine.org/User_name_and_password_not_remembered

Note that Firefox won't auto-fill saved form data automatically when autocomplete=off is used, so you need to double-click the empty name field to get a drop-down list.

I'm sorry but this is not the problem. The old login page still autocompletes and opens the website, but causes problems once the site is open. When I check Mozilla it tells me that I already have a password stored so doesn't offer to save another.

Thanks

You should be prompted to update your password, if you're typing in a different one than the one that's currently stored.
You should also be prompted to save login information if you're logging in with a different user name than the one currently stored; this wouldn't replace the existing information.

I suppose if the site design changed, it's possible something has gone wrong. Try removing the remembered login:

• Password manager - Remember, delete and change saved passwords in Firefox

But if you're still not prompted to save the login information, the site preventing that is still the most likely cause. And for that, the simplest solution is to install the aforementioned Remember Pass add-on.
If you don't want to post a link to the login page, you can examine it yourself. Right-click the user name field and choose Inspect Element. Check for the presence of the autocomplete=off attribute for either the input elements, or the parent form element.