Ideas to include
Their entire personality
The bark at the mail carrier, the one toy, the spot on the bed that was legally theirs. This is what makes it sound like your animal and not a dog in general.
How they arrived
The shelter, the litter, the fact that they chose you and not the other way round. Start there and the whole arc is already in place.
For the kids
A first loss, and it is a real one. A song gives a family something to do together instead of everyone being sad in separate rooms.