That's not the same, though.
Mockingjay is the title of the third book of what's called The Hunger Games trilogy.
The movies retain the titles of the book.
This isn't a retroactive decision. Check out this boxed set of the trilogy which was released in 2010 - two years before even the first movie came into the theaters: https://www.amazon.com/dp/0545265355
The new "Lord of the Rings" TV series, however, is not part of the Lord of the Rings trilogy*.
Should they keep the LotR label, then it will undoubtedly be because of brand recognition, as you imply. But it won't be the same as the Mockingjay example.
A better analogy would be to call the Fantastic Beasts and Where to Find Them films "Harry Potter". Only instead of the time gap being two generations it would be two millennia (or so).
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* I'm calling it trilogy for the sake of this discussion; I'm aware that it isn't supposed to be a trilogy
When you buy the hardcover box set that is exactly what they look like without the dust jackets.
Link in the blog is a referral link, FYI. Here is a non-referral link:
http://www.amazon.com/The-Hunger-Games-Trilogy-Boxed/dp/0545265355/