Having a pet provides different advantages to different people. Companionship is at the top of the list. To an elderly person this extends much more than one can sometimes imagine.
Many older people live by themselves.
Many more live away from family and friends. Isolation is one of the worst downfalls of getting older in our society. Bringing a pet into the household sounds like a simple way to fix much of this, almost too simple. Make sure they want and can care for the pet. The pet will need daily support in feeding, cleaning, keeping healthy, and even playing.