What does the "s" in "thanks" mean?

Solution 1:

You're right that "thanks" is always used as a plural noun, whereas when we say "thank you," the word "thank" is a verb.

You can explain that some nouns are always used in their plural form, and give other examples (clothes, outskirts, glasses, scissors, etc.).

You could alternatively tell your students each expression of thanks is a shortened form of a longer sentence that isn't really in use anymore.

(I) thank you. -- Thank you.

Thanks (be to you). -- Thanks.

or

(I give/feel) thanks. -- Thanks.

Solution 2:

My interpretation is that my is elided.

[My] thanks!

A longer interpretation would be:

[I give you my] thanks!

You could also say that it's imperative:

[Take/receive my] thanks!

Thanks is one of many responses we use when time is short or a longer expression seems egregious or pretentious. In a similar vein, we respond to good news with congratulations! an unintended slight with apologies! or sad news with condolences.