Tamagotchi
A lot of people would remember this one due to how popular it was and the different versions it had. You are basically raising a pet from it's hatching, into feeding and caring for it through growth up to adults and ultimately, up until (if it wasn't fed or it was confiscated at school for the day) its death.
You can get a wide variety of Tamagotchi's based on different sorts of animals which I cannot remember exactly what my one was as a kid, but I loved that thing, getting a lot of time use from it. Then they have different sorts including a Digimon battle sort of device with a similar concept. I never got to battle mine too much but it was fun none the less and I was a happy kid.
Kids today can still benefit from looking after one, especially with early use of mobile phones and tablets from a young age, but provide a sense of caring and responsibility. Although they should be maybe five years old and up to better understand what to do and how to care for it.
There are so many memories from my childhood that I miss and wish I could relive as everything in this day and age isn't so simple with the technology being a part of nearly everything, the way the world is and the way people act.