Sorry for your loss. It seems that you will carry on where the boys father left off - all the best for everything that comes with that responsibility.
+1 in regards to the maturity of the kid. My oldest was almost 6 when he put the first .22's downrange with a Crickett. Before that we shot lots of airsoft pellets in the garage out of an Airsoft M4 (collapsible stock all the way forward is about the right LOP).
My younger son is now almost 7 is starting to develop an interest in shooting. When I started looking around for a rifle for him, I came across the Savage Rascal
. Same size as the Crickett, but with a significantly better trigger. According to a Lady in their CS dept. the first ones are scheduled to ship early March.
My older son meanwhile has developed an interest in pistols. The Walther P22 fits his hand perfectly.
Edited to add: Reactive Targets rule! We used large plastic coffee cans, filled with water and frozen overnight - every hit sends a fountain of ice flying