The tty command doesn't really tell you which terminal you're using, rather it tells you which terminal port on the PDP-11 you're using. So restarting your terminal won't change what tty outputs, but swapping cables with an adjacent terminal would (or alternatively, swapping which ports the terminal cables connect to).
You can change the name of the port by renaming the appropriate special file in the /dev directory. If you do this you will also need to update the /etc/ttys file which lists the names of the ports that users may use for logins.
Note that pseudo-terminal (pty) names are not fixed, because you will get assigned a pty from a pool of available ptys when you connect. So if you connect to the system over the network using telnet, you might be assigned ttyp0, or ttyp1, or ttyp2, etc., depending on which ones are in use already. But if you're the only person who uses the computer then you can be pretty sure you will always end up being assigned ttyp0.