I'm trying to clean up notifications for a client site and was wondering if the notification pattern that Stack Exchange employs has a name that I can use to search for more examples.

I'm trying to clean up notifications for a client site and was wondering if the notification pattern that Stack Exchange employs has a name that I can use to search for more examples.

The terms I see the most are "message bar", "notify bar" and "notification bar".
There are tons of examples with Google using the terms above. Below are a few.
Top Floating message box using jQuery
jQuery: How to show a message bar at the top of browser window
Within Google, we call it a Butter Bar, to disambiguate it from all the other types of notifications that might also be shown on a page.
I just found a good page from Chromium project about this UI element. They call it infobar and have there quite nice guidelines, rationale and examples: http://www.chromium.org/user-experience/infobars
As Rahul mentioned, this is called the "flash" in Ruby on Rails, CakePHP, Yii, Lithium and TYPO3 at least.
It's called banner in material design: https://material.io/design/components/banners.html