I just tried and was going to file a ticket when I found this one. Here's what I found:
I changed the name of a repo, and afterward the Overview, Source and Changesets tabs no longer show contents -- they look as if it's a brand new repo. However, associated followers and patch queues *do* seem to stay related. I haven't tested forks. Also, trying to clone from the new URL after the name change did not work (well, technically it did, but I got an empty repo with no change history).
Changing the name back to what it was before sets everything right again.
Yep, actually when renaming a repo, the directory stays intact on disk. It needs to be renamed together with the repository. I'll need to think about whether this is eligible, or whether not allowing repository renames is a better way to go..
If it's as simple as a rename, that's fine, but forks use hardlinks, so. Well, I'll look more into it.