Yes, I think she should. Eight is old enough to take on the sort of responsibilities involved in looking after a pet. Under a certain amount of supervision, of course, as you can't let the pet suffer. Always check that it is fed and has fresh water and that the litter tray has been cleaned if the cat is kept in at night. Don't make it too obvious that you are checking these things, you can always make it look as if you are taking an interest in the child and are proud that she has taken on the responsibility.
I think that learning how to care for another living being is a wonderful life lesson, and one that will stand any child in good stead later in life.