The confusing Unix permissions terminology The GNU manpage (what you'll find on a Linux system) – but pretty much any Unix manpage – of chown (the command for changing the owning user and owning group in Unix) appeal to an “owner” and, as you see, the very name of this command hints at  “change owner”, too.

When people first learn this, they get the wrong idea, that for the command chmod (which sets the file/directory permissions), the abbreviation o (as in chmod o+w file) stands for “owner / owning user”. 

In reality it stands for “other users”. So chmod o+w file really adds the writing permission (+w) for other users (who are not the owner or a member of the owning group). It doesn't help that what really refers to the “owner”, namely u (= owning user), makes it possible for one confusion to give rise to another: u is then confused for “‘normal’ users”: anybody not the owner or in the owning group.

Sure, it helps to make people directly aware of this, but even if they can remind themselves that there's some trap here, the idea o = “owner”, u = “normal users” is so seductive, that they retain some insecurity until enough (sometimes a lot is necessary) rote memorization makes it finally sink in.

What would be a mnemonic or other way to explain it which makes the correct notion stick right from the start, so rote memorization isn't necessary?

PS: I'm not formally teaching anyone, it's just what I want to tell coworkers/friends. When I took a university course on Linux I first came across this confusion. There's a lot more about the Unix permission system which is very counter-intuitive, but those are issues arising necessarily from technical aspects. Not an (surprisingly very) annoying and trivial issue caused by bad historical choices for abbreviations.
You could point out that a file has an owning user and an owning group. Since “owner” appears twice, specifying o for “owner” would be ambiguous. That's weak because the owning group doesn't really have any special privileges on the file, unlike the owning user: the owning user can change the file's permissions and other metadata, but the owning group only has the privileges granted by the file's permissions.

I think the best approach is to pair up user and group so that people remember them together. There's user and group: u and g. There are basic file permissions for one user and for one group. ACL entries can apply to either a user or a group. The command ls -l has two columns for ownership, one with the user and one with the group. Beware that sometimes the naming is not consistent. On the system there's a user database (/etc/passwd) and a group database (/etc/group). The chown command takes both a user and a group, but there's also a separate chgrp command and so chown is often used for the user only, which tends to reinforce the association between “user” and “owner”, so avoid mentioning that if you're trying to dispel the association between “user” and “owner”.