It's tough. Been through it myself, kids about the same age. Waited about 16 months for the breeder to get a new pup. The new dog ain't the old dog. Both are special to me, but the oldest misses our old dog a lot. After almost 3 years though that's better. The youngest adored them both.
The oldest was very concerned about forgetting the old dog, that was an interesting concern that was new for me as an adult. She still speaks fondly of him.
Let your child be sad and comfort them. Show them your emotions in this too. Hug and cry together and just walk through it with them, don't try to solve it for them. One because you can't, two because it's a healthier way for them to process on their own, imo.
When the family is ready, pick the dog for the family by your desires. Kids/spouses tend to sway opinions about which dog to bring home. I hunt my dogs so I'm looking for specific traits. The family often would simply be happy for a life support system with a tail.
One thing I did do was fun. I surprised then with the new pup with the assistance of my wife. I also showed them pics from the breeder of our pup telling them this was like the pup I wanted to get and gave them some choices for names as we looked at pictures. They still laugh about seeing him in photos anf the collective, "Oh that's cute", that inevitably followed. When we settled on a name we started using it before they even knew we were getting a pup.
Time is a healer.
Time as noted is the healer and when the time arrives a puppy will create the new bond
