Yes, definitely teach her how to drive a manual. I taught our daughter about 7 years ago. Since then her travels have taken her many places in the world where the only rental cars available were manual transmissions. And several times she was the only one in her group who knew how to drive a manual.
When she was learning how to drive a manual, she had no clue why I was encouraging her to do so. Since then she has said several times how glad she is that she knows how.
Also, here’s something I’ve posted before. Years ago a father asked if he should teach his daughter how to drive a standard shift. Someone replied with this, which hit home for me because it was the same experience I had with our daughter:
Teach her yourself. On a quiet Sunday morning, drive to a big parking
lot by some circuitous route where you’re showing her the basics of
shifting. Then let her try. When she lets the clutch out for the first
time, the car will stall. She will immediately look at you for help
using the same smile she used when she was 6 and you knew everything in
the universe.
Remember that precious moment because it may be the last time you’ll experience it.