I've had a request to change the name of a Jira project.
Seems simple enough but is there any possible issues that would generate? There are associated issues with the project so I'm worried about the key. Should that stay the same?
Changing the Project Name does not automatically affect the Project Key. You can change the name without changing the Key.
I generally advise against changing the Key unless there is a truly compelling reason to change it. If you do change the key you should note that the original key cannot be re-used for a different project.
No, you won't hit problems, unless you have configured something that looks up the project by name (which is generally a bad idea, the key is a better way to do it, or ideally the project ID, as that can't change)
The project's name is a simple attribute, stored when you create the project, and looked up by any code that wants to display it. If you edit it, the code that wants to display it will re-read it when it needs it. Nothing else really uses it.