Leaving money to a pet depends on what state you live in. Some states do allow pets to receive money in what's called a pet trust. However, it's best to simply make arrangements before you die for the care of any pets that outlive you.
You can contact your local humane society, most of these societies have programs that allow you to donate to the society through the will or through the trust and make provisions for a pet.
Or you can simply leave a small cash gift to a friend who agrees to take care of the pet.