But the GPL grants rights to use the code, which itself contains the name. Is there language in the license that exempts "the source code that contains the project name" from my ability to copy and redistribute it?

You're asking the wrong questions because you're mixing up copyright and trademark law.

The GPL covers copyright. It includes the source code, which as you noted contains the project name. This is why you may click "fork" on GitHub and wind up with a fork that is "yours" under the original name.

Trademark law is usage based. If you then set up a website for your fork under the original name, that's illegal. The fact that you're allowed to have a fork containing that name isn't relevant. Copyright allows the code, but you're now violating trademark law.

If you copy and redistribute the code as-is, on a website called, let's say opensourcesoftware.com, nobody can stop you.

If you copy and redistribute the code on a website called thisistheofficalsitefornotepadplusplus.com, because the ownership of the name Notepad++ is separate from the code.

If you copy and change the code before redistributing it, they can say you're not allowed to use the name Notepad++ and must change that part of the code. You asked a very pertinent question with "Does the GPL say anything at all about trademarks?" It does not, because it has nothing to do with copyright. It does say if you modify the source code, you must make available the your modifications under the same license.

If it helps, here's another way to think about it:

If you're a photographer, and someone signs a photo release form for you, that means you're allowed to use their photo in marketing materials.

It doesn't mean you're allowed to use their photo to get a passport in their name.

The photo release form doesn't need to have additional language saying "the photographer agrees not to use the photo to create a fraudulent passport" for this to be disallowed.