Notification, sudden error: Warning: include(): Filename cannot be empty

Unfortunately, there's not enough information in your question to accurately debug this, and based on the specifics of your call stack, it looks like this system has received a few customizations which may be interfering with standard store behavior. Here's some observations that may help you track down the problem

1. Magento's trying to include a template block with an empty string set as it's template. That's why you're seeing the warning. Look for any new (or old) setTemplate, or template="" in your code, or any place a template block is instantiated without setting a template

2. Magento's trying to render the template system/cache/notifications.phtml — however, the fact its fallen back to the base design package to do this means Magento couldn't find this file in any adminhtml package/theme. This is a stock file that ships with the system, and may have been removed

3. You say you're seeing it "everywhere", but the errors indicate it's an admin only problem. This might mean you're trying to run admin code from the front-end of vice versa (again, hard to remotely debug this sort of thing)

See How can I debug "empty template": Filename cannot be empty in Template.php

We solved this by adding

    if (strpos($includeFilePath, realpath($this->_viewDir)) === 0 || $this->_getAllowSymlinks()) {
        if (empty($includeFilePath)) {
            Mage::Log("Cannot set emptt filename" . $fileName . " on a " . get_class($this) );
            Mage::log(Mage::printDebugBacktrace(), null, 'backtrace.log'); //or you can even log the backtrace
            include $includeFilePath;
        } else {
            include $includeFilePath;
        }
    } else {
        Mage::log('Not valid template file:'.$fileName, Zend_Log::CRIT, null, null, true);
    }

where 'Mage::printDebugBacktrace()' was grabbed from here

http://www.blog.magepsycho.com/utilizing-debug_backtrace-function-for-magento-debugging/

This way we had some more info to go by ...