After accidentally renaming /usr, how do I rename it back?

Since you have set a password for root, use su and busybox, installed by default in Ubuntu. All of su's required libraries are in /lib. Busybox is a collection of utilities that's statically linked, so missing libraries shouldn't be a problem. Do:

su -c '/bin/busybox mv /usr_bak /usr'

(While Busybox itself also has a su applet, the /bin/busybox binary is not setuid and so doesn't work unless ran as root.)

If you don't have a root password, you could probably use Gilles' solution here using LD_LIBRARY_PATH, or(Gilles says this won't work with setuid binaries like sudo) reboot and edit the GRUB menu to boot with init=/bin/busybox as a kernel parameter and move the folder back.

In addition to muru's answer:

• you could have used some rescue boot USB key to repair your system; e.g. if your system is some Debian or Ubuntu, boot the installation USB key in rescue mode, and do the appropriate mount and mv and umount.

• to be able to repair more easily such mistakes, I generally also install a static shell with several builtin commands (notably with some cp, rm, mv-like builtins) such as sash (it is packaged in Debian & Ubuntu, and also available as sash-3.8.tar.gz in source form) and boot with init=/bin/sash passed to Grub.

^{PS: sash is slightly buggy, and not entirely Posix compliant, but still very useful.}

I think the best safest way is to reboot using a USB, CD or DVD booted OS (Debian, Ubuntu, Suse, etc). Then mount the drive containing the problems and do the rename.

Safer than booting into a minefield with /usr or /lib effectively missing.