Is it okay to use both, give and teach, in this expression :
The next time she's late, go without her. That should teach her a lesson.
The next time she's late, go without her. That should give her a lesson.
In BrE, "That should give her a lesson" would sound slightly unidiomatic; though the meaning would probably be clear.
"That should teach her a lesson" is fine; though it does always sound almost as if the speaker takes pleasure in the thought of the lesson that is to be taught!
All the best,
Not a professional ESL teacher.