Seeing as TI83+ OS recieving has been cracked, it would be quite easy to burn someone's name into the OS, maybe replace the string Ram Cleared with someone's name, or some other text string, or even add a new one in there.

Yes, but it wouldn't be legal to distribute such an OS, now, would it?

I really like the idea of having very-difficult-to-remove user information. (This will definitely be a feature of any actual usable OS I ever get around to writing.)

But unfortunately, whether it's placed in the OS itself (by modifying the RAM clear strings, or similar) or whether it's placed somewhere else and the OS is hacked so as to display it, you can undo the modification simply by sending a new OS. (Unless of course you want to open it up, unprotect the chip, and flash the boot code.) So to be really effective it would need TI's cooperation.

Which is not to say that it's a bad idea anyway...